Systems and method for assigning credit and for generation with credit-assignment for music and other sequence data

Abstract

A network-free framework for machine learning incorporates generative modeling by generalizing a parameter-free, statistically consistent data interpolation method. Embodiments can be applied to various types of sequence data (e.g., audio / music, image, video, text, numerical, financial, medical, etc.) as well as to various types of outputs (e.g., assigning credit, generating predictions, compensating contributors to training data, sequence generation with provenance, sequential data compression / decompression, feedback control, and reinforcement learning.

Description

[0001] SYSTEMS AND METHOD FOR ASSIGNING CREDIT AND FOR GENERATION WITH CREDIT-ASSIGNMENT FOR MUSIC AND OTHER SEQUENCE DATA

[0002] CROSS-REFERENCE TO RELATED APPLICATION(S) This patent application claims the benefit of United States Provisional Patent Application No. 63 / 906,719 entitled SYSTEMS AND METHOD FOR ASSIGNING CREDIT AND FOR GENERATION WITH CREDIT-ASSIGNMENT FOR MUSIC AND OTHER SEQUENCE DATA filed October 28, 2025 and the benefit of United States Provisional Patent Application No. 63 / 712,841 entitled ARTIFICIAL INTELLIGENCE WITHOUT NETWORKS filed October 28, 2024, each of which is hereby incorporated herein by reference in its entirety.

[0003] FIELD OF THE INVENTION

[0004] The invention generally relates to the fields of artificial intelligence (Al) and machine learning, and, more particularly, to systems and method for assigning credit and for generation with credit-assignment for music and other sequence data.

[0005] BACKGROUND OF THE INVENTION

[0006] Contemporary Artificial Intelligence (Al) stands on two legs: large training data corpora and many-parameter artificial neural networks (ANNs). The data corpora are needed to represent the complexity and heterogeneity of the world. The role of the networks is less transparent due to the obscure dependence of the network parameters and outputs on the training data and inputs. This raises problems, ranging from technical-scientific to legal-ethical.

[0007] SUMMARY OF VARIOUS EMBODIMENTS

[0008] In accordance with one embodiment of the invention, a system and method for assigning credit to artificial intelligence training items involves receiving a list of a plurality of training items and a new item of a same type as the plurality of training items, associating a numerical vector of a pre-determined dimension to each training item as well as to the new item, wherein the pre-determined dimension is the same for each training item and the new item, computing, for each training data item, a non-negative weight quantifying resemblance of the new item to the training data item, where such weights are specified by a predetermined function of training data vectors and the new vector, and which does not involve any optimization or tunable parameters at any time during the credit assignment process, and assigning credit to at least one of the training items based on the weights.

[0009] In accordance with another embodiment of the invention, a system and method for generating predictions involves receiving a list of training input and output pairs, and a new input, for which an output prediction is desired, associating numerical vectors of the same pre-determined dimension to each training input item as well as the new input item, computing a non-negative weight for each training input quantifying its resemblance to the new input, where such weights are specified by a predetermined function of training data vectors and the new vector, and which does not involve any optimization or tunable parameters, and computing the desired prediction by combining the training outputs and their associated weights.

[0010] In accordance with another embodiment of the invention, a system and method for compensating contributors to training data for generative Al involves receiving a list of training items and a new item of the same type, associating a numerical vector of the same pre-determined dimension to each training item as well as the new item, computing non-negative weights quantifying the resemblance of the new item to each training data item, where such weights are specified by a predetermined function of the training data vectors and the new vector, and which does not involve any optimization or tunable parameters at any time during the process, and assigning compensation proportionally to these weights.

[0011] In accordance with another embodiment of the invention, a system and method for sequence generation with provenance involves receiving a set of training input sequences, creating a set of training input-output pairs, where the training inputs are fragments of the input sequence of fixed length T, and the training output for each such fragment is the sequence element immediately following the said fragment, associating numerical vectors of the same pre-determined dimension to each training input item, initiating the sequence generation process with a starting fragment of the same length, computing a non-negative weight for each training input quantifying its resemblance to the new input, where such weights are specified by a predetermined function of training data vectors and the new vector, and which does not involve any optimization or tunable parameters, generating a sequence element following the starting fragment by combining the training outputs and their associated weights, associating with the generated element, the set of weights for individual training data fragments used to generate the said element, and defining the fractional provenance of the generated element in terms of the individual training data fragments to be given by the said weights, generating a new starting fragment by dropping the earliest element of the previous starting fragment, and appending the newly generated element from the previous step, and iterating the process of generating subsequent elements.

[0012] In accordance with another embodiment of the invention, a system and method for sequential data compression involves receiving an input sequence, fixing a subsequence length T, which length is fixed in advance, or user selected for the sequence in hand, creating a set of time-ordered training input-output pairs, where the training inputs are frames consisting of consecutive elements of the input sequence of fixed length T, and the training output for each frame is the sequence element immediately following the said fragment, storing the first frame and the sequence element immediately following that frame in a data store for holding the compressed sequence, shifting the frame forward by one time point, and predicting the next time point using the described methods, with the training data inputoutput pairs consisting of frames preceding the current frame and their immediately following sequence elements, computing the residual between the predicted sequence element following the current frame, and the actual sequence element, and storing that residual in the data store for holding the compressed data, and repeating the above two steps until the input sequence is exhausted.

[0013] In accordance with another embodiment of the invention, a system and method for decompressing the stored compressed data generated by the above compression scheme involves receiving a compressed data store generated as above, initiating the decompressed sequence using the stored first frame and the stored sequence element following that frame, for each subsequent sequence element to be appended to the decompressed, computing the predicted sequence element using the decompressed sequence up to the current position, as the source of training input-output pairs, retrieving the stored residual for the predicted sequence element, and adding that residual to the predicted sequence element to obtain the decompressed sequence element, and repeating the two above steps until the compressed data store is exhausted.

[0014] In accordance with another embodiment of the invention, a system and method for feedback control involves receiving an initial pair of training sequences of measurements constituting system outputs, the corresponding sequence of controlling inputs, and a cost function that takes one or more values of the measurements and the values of the corresponding controlling inputs, and produces a positive cost as an output, using the above method with the sequence of past measurements and control values as inputs and the measurements at subsequent time points as outputs, to predict the system output at the next time step for a hypothetical control input at the present time, and computing the predicted resulting cost using the predicted output and the hypothetical control input, iteratively repeating the preceding step of predicting output for hypothetical control inputs, for a fixed time horizon into the future comprising one or more time steps, and computing and adding the corresponding predicted costs to obtain the total predicted cost for that sequence of hypothetical control inputs, repeating the procedure in the previous two steps with different choices of the sequences of hypothetical control inputs, and subsequently choosing that optimal sequence of hypothetical control inputs in which minimizes the total predicted cost, executing the first control input of that optimal sequence, and receiving the subsequent output measurement, and adding the new control input and the subsequent measured output to the training data sequences, and iterating the process described in the previous steps.

[0015] In accordance with another embodiment of the invention, a system and method for reinforcement learning involves receiving an initial pair of training sequences of measurements constituting system outputs, the corresponding sequence of controlling inputs, and corresponding reward values, using the method of claim 48, with the sequence of past measurements, rewards and control values as inputs and the measurement and rewards at subsequent time points as outputs, to predict the system output as well as the predicted reward value at the next time step for a hypothetical control input at the present time, iteratively repeating the preceding step of predicting outputs and rewards for hypothetical control inputs, for a fixed time horizon into the future comprising one or more time steps, and adding the corresponding rewards to obtain the total predicted reward for that sequence of hypothetical control inputs, repeating the procedure in the previous two steps with different choices of the sequences of hypothetical control inputs, and subsequently choosing that optimal sequence of hypothetical control inputs in which minimizes the total predicted reward, executing the first control input of that optimal sequence, and receiving the subsequent output measurement as well as reward value, and adding the new control input and the subsequent measured output and reward value to the training data sequences, and iterating the process described in the previous steps. Additional embodiments may be disclosed and claimed.

[0016] BRIEF DESCRIPTION OF THE DRAWINGS

[0017] Those skilled in the art should more fully appreciate advantages of various embodiments of the invention from the following “Description of Illustrative Embodiments,” discussed with reference to the drawings summarized immediately below.

[0018] FIG. 1 is an illustration of the Hilbert interpolation scheme in one dimension.

[0019] FIG. 2 shows modular replacement of ANN parametric function module with Hilbert kernel module.

[0020] FIG. 3 shows relevant fish variables and flow chart of the Hilbert interpolation scheme.

[0021] FIG. 4 shows a comparison of the behavior of real and Hilbert fish for N= 2 individuals.

[0022] FIG. 5 shows a comparison of the behavior of real and Hilbert fish for N= 5 individuals.

[0023] FIG. 6 shows input-output prediction using Hilbert autoregression (Eq. (10)).

[0024] FIG. 7 shows Hilbert Predictive Control of nonlinear dynamical systems.

[0025] FIG. 8 shows predicting zeros of the Riemann function.

[0026] FIG. 9 shows correlations of the zeros of the Riemann function.

[0027] FIG. 10 shows an example of classification using the Hilbert kernel.

[0028] FIG. 11 shows an example behavior of 2 Hilbert fish without the tank wall.

[0029] FIG. 12 shows an entropy time series for 2 Hilbert fish.

[0030] FIG. 13 shows an entropy distribution for 2 Hilbert fish.

[0031] FIG. 14 shows an example of autoregressive generative modeling of time series data. FIG. 15 shows another example of autoregressive generative modeling of time series data.

[0032] FIG. 16 shows the impact of an initial coordinate transformation x —> <b(x).

[0033] It should be noted that the foregoing figures and the elements depicted therein are not necessarily drawn to consistent scale or to any scale. Unless the context otherwise suggests, like elements are indicated by like numerals. The drawings are primarily for illustrative purposes and are not intended to limit the scope of the inventive subject matter described herein. DESCRIPTION OF ILLUSTRATIVE EMBODIMENTS Various embodiments are described in United States Provisional Patent Application No 63 / 712,841 entitled ARTIFICIAL INTELLIGENCE WITHOUT NETWORKS filed October 28, 2024, which is hereby incorporated herein by reference in its entirety and forms part of the disclosure of this patent application, including the following Appendices which were incorporated physically and by reference into the provisional patent application:

[0034] (1) Parameter-free Statistically Consistent Interpolation: Dimension-independent Convergence Rates for Hilbert kernel regression, published on or about June 7, 2021 (Appendix A);

[0035] (2) Al Without Networks, published on or about December 9, 2023 (Appendix B); and

[0036] (3) Al Without Networks, published on or about June 6, 2024 (Appendix C).

[0037] The disclosure below is substantially derived from the “Al Without Networks” publications for both physically incorporating relevant material into the patent application and expanding certain concepts and claims. Certain embodiments are described as examples, which should not be construed as limiting.

[0038] Introduction

[0039] It should be noted that the term “Hilbert kernel” is used herein to denote the weights utilized in our approach. This term was first utilized for such weights in work by Devroye et al as cited in the attached papers. Despite our utilization of this original nomenclature here, we go well beyond what was originally being done with this kernel, both in terms of the generative methodology which requires additional algorithms and the different specific developments beyond the generative methodology as required by the different domainspecific applications. It also should be noted that the Hilbert kernel is referred to as the Shepard kernel in the computer graphics and data interpolation literature which has largely developed independently of AI / ML and pre-dates it. However, this literature largely does not concern itself with statistical function learning issues that are central to modern AI / ML and this kernel has generally not found usage in modern AI / ML developments. It should be noted that certain embodiments are described with reference to various types of sequence data, which, for the sake of this disclosure and related claims, are fragments or information of a given type such as audio (e.g., music), image, video, text (e.g., word tokens), numerical, financial, medical (e.g., biomedical), or any other type of sequence data.

[0040] Contemporary Machine Learning (ML) stands on two legs: large training data corpora and many-parameter artificial neural networks (ANNs). The data corpora are needed to represent the complexity of the world. The role of the networks is less transparent due to the obscure dependence of the network parameters and outputs on the training data and inputs. This raises problems, ranging from technical-scientific to legal-ethical. We hypothesize that a transparent approach to machine learning is possible without using networks or parameters at all. By generalizing a parameter-free, statistically consistent data interpolation method, which we analyze theoretically in detail, we develop a network-free framework for ML incorporating generative modeling. We demonstrate this framework with examples from three different disciplines - ethology, control theory, and mathematics. Our generative Hilbert framework applied to the trajectories of small groups of swimming fish outperformed state-of-the-art traditional mathematical behavioral models and current ANN-based models. We demonstrate pure data interpolation based control by stabilizing an inverted pendulum and a driven logistic map around unstable fixed points. Finally, we present a mathematical application by predicting zeros of the Riemann Zeta function, achieving comparable performance as a transformer network. We do not suggest that the proposed framework will always outperform networks as over-parameterized networks can interpolate. However, our framework is theoretically sound, transparent, deterministic, and parameter free: remarkably, it does not require any compute-expensive training, does not involve optimization, has no model selection, and is easily reproduced and ported. We also propose an easily computed method of credit assignment based on this framework, to help address ethical-legal challenges raised by generative Al.

[0041] Contemporary ML continues to have a large impact across many domains of activity [1], encompassing technological, scientific, and creative domains. While the applications are diverse in computational architecture, they are all based on two basic elements: (i) functions approximated by artificial neural networks, which are nonlinearly parameterized by many parameters [2], and (ii) large sets of data relevant to the application domain, collected from the real world [3] or simulated computationally [4], Although these applications have been pragmatically successful in specific domains due to high performance in certain complex tasks [4-6], there are significant challenges [7], These stem from the lack of transparency and interpretability of the networks [8], particularly important in scientific [9] or biomedical applications [10-12], as well as in ethical-legal issues related to authorship and ownership of the output of generative Al

[0013] , Other related problems are difficulties in reproducing results and porting code

[0014] , due to stochastic elements in training and sensitivity to network details.

[0042] These challenges arise fundamentally from the non-transparent, nonlinear relation between the data corpora and the network parameters, determined using various online and offline optimization procedures (learning algorithms). The centrality of this parameter optimization step is clear - the overall subject area is called machine learning. Additionally, the multi-layer nonlinear parameterization of the functions involved means that the inputoutput relationships of the optimized (trained) networks are also nontransparent. This has led to attempts to develop interpretable network architectures

[0015] , but ANNs are complex nonlinear parameterizations and will generally not bear a transparent relation to the training data. One possible solution is offered by recent work [16-18] showing that the networks may be acting as data interpolation devices, thus prioritizing the data corpora over the networks. However, recent work in this area has focused on conditions under which the data interpolation provides good performance (“benign interpolation”

[0019] ), rather than solving the issues above. We suggest that a fundamentally transparent approach to machine intelligence is possible for sufficiently large data corpora that eliminates the networks entirely, thus cutting the Gordian knot of interpreting the network parameters and eliminating conceptual confusions that may arise from the complexity of the networks. This radically empirical approach - all data and no model - also automatically solves the problem of understanding network outputs in terms of the inputs and facilitates reproducibility and portability.

[0043] The basis of our approach is an interpolation methodology, which remarkably has no tunable parameters. The Hilbert kernel interpolator

[0020] , a classical NadarayaWatson

[0021] regression scheme using a singular weight function, was previously shown to be statistically consistent

[0020] (producing Bayes optimal results for large data sets), but has not been widely used. We show that this interpolator belongs to a large class of interpolation methods, which we term the generalized Hilbert kernel. We perform a rigorous and detailed theoretical analysis, including a closed form analytical expression for the asymptotic convergence rate, which we show is strikingly independent of data dimension. We propose a general framework, based on the generalized Hilbert kernel and inclusive of generative modeling, and apply it to problems of a scientific and theoretical nature from three disciplines, ethology, control theory, and mathematics. In a study of collective animal behavior, the framework applied to groups of 2 and 5 rummy-nose tetra fish outperforms both the current best data-driven behavioral models [22-24] and ANN models [25, 26] for this species. In a control theory example, we replace the system model normally used in a model-predictive control (MPC) [27, 28] framework with a generalized Hilbert predictor, to control an inverted pendulum as well as a chaotic map. Finally, the framework is used to predict zeros of the Riemann zeta function, with similar performance as a transformer trained on the same task, both predictors having presumably reached the Bayes limit.

[0044] While we demonstrate good performance in these applications compared with ANNs, this is not our main take-home message, as performance is not the only criterion with which to evaluate such an approach, particularly in scientific and biomedical applications where understandability, transparency, and theoretical grounding is of great importance.

[0045] Overparameterized networks can interpolate and may also provide good performance. There is however a major difference: the Hilbert approach involves no learning, in contrast with all network-based Al. The Hilbert framework does not have the issues of lack of transparency, reproducibility, and is able to easily perform credit assignment, while demonstrating performance in scientifically sound settings. The framework is theoretically well-grounded, deterministic and parameter free: it does not require compute-expensive training, does not involve stochastic optimization, does not include model selection, and is easy to reproduce. These are critical considerations for scientific applications of machine learning, which we consider our primary target in this work, although we expect that the utility of the approach will extend to engineering applications as well, particularly given legal issues surrounding copyright and data provenance. The Hilbert generative framework also shows interesting phenomena such as copying portions of training data, which have been recently observed in the context of Large Language Models

[0029] , and provides a theoretical basis for understanding such behavior.

[0046] We first discuss Statistically Consistent Interpolation (SCI) and present a summary of our analytical and rigorous results for the behavior of the Hilbert kernel interpolator, together with numerical illustrations of selected results. Next, we propose a framework for interpolative generative and system modeling of scalar or vector time series, using the generalized Hilbert kernel. We present three applications of this framework. The first application is to the collective behavior of groups of 2 and 5 rummy-nose tetra fish, for an in-depth demonstration of utility and performance in the wide context of collective animal phenomena. The second application is to control theory and engineering, where we demonstrate performance in a canonical example, the stabilization of an inverted pendulum, using a Hilbert predictor to replace the system model. The third application is to a mathematics problem, where we exploit the Hilbert scheme to predict the zeros of the Riemann zeta function, and compare its performance to that of a transformer network trained on the same task.

[0047] We discuss the ramifications of our results, for the applications addressed here, as well as in more general contexts. Finally, we propose an approach for credit assignment that helps address ethical-legal challenges raised by generative Al and has implications for copyrighted training data, and discuss other ramifications of a network-free, data interpolation-based transparent approach to Al.

[0048] Results and discussion

[0049] Statistically consistent interpolation and the Hilbert kernel

[0050] Data interpolation and statistical regression are both classical subjects but have been largely disjoint until recently. Scattered data interpolation techniques

[0030] are generally used for clean data. On the other hand, when supervised learning or statistical regression techniques are applied to noisy data, smoothing or regularization methods are usually applied to prevent training data interpolation, as the latter is believed to lead to poor generalization

[0031] , However, empirical evidence from overparameterized deep networks has shown that data interpolation (equivalently, zero error on the training set) does not automatically imply poor generalization [32, 33], This has in turn given rise to much theoretical work to understand how and why noisy data interpolation can still lead to good generalization [16, 34-43], Note, however, that this body of work is largely within the framework of parameterized networks, linear or nonlinear, and generally involves a model fitting / selection step, in contrast with our network and parameter-free approach. A key notion in this regard is the phenomenon of statistically consistent interpolation (SCI)

[0018] , i.e., regression function estimation that interpolates training data but also generalizes as well as possible in the sense of statistical consistency by asymptotically achieving the Bayes limit for expected generalization error (risk) when the sample size becomes large. The Hilbert kernel

[0020] is a Nadaraya-Watson style estimator [21, 44] with the unique property that it is fully parameter-free, as it does not have any bandwidth or scale parameter. It is global and uses all or batches of the data points for each estimate. The associated kernel is a power law and thus scale-free. Although statistical consistency of this estimator for ^ / -dimensional input data was proven when it was proposed

[0020] , there has been no systematic analysis of the associated convergence rates and asymptotic finite sample behavior, and it is generally not used in applications. We show that the Hilbert kernel is a member of a large class of SCI methods that have the same asymptotic convergence rates.

[0051] The only other interpolation schemes we are aware of, that were proven to be statistically consistent in fixed dimensions under general conditions, are NW-style schemes using singular weighting functions. To the best of our knowledge, the first published proposal along these lines was the weighted interpolating nearest neighbors method (wiNN)

[0016] (see section 4.1 and 4.2) with min-max optimal convergence rates. There are several follow up works in the literature

[0045] ,

[0046] examining NW interpolation with singular kernels. Note that

[0045] utilizes the same power law weights as wiNN but with a fixed spatial bandwidth rather than a fixed number of nearest neighbors. This is not a significant conceptual difference as fixing the number of neighbors and the spatial bandwidth lead to the same asymptotic results for a large number of samples.

[0052] While wiNN and related singular NW kernel methods superficially resemble the Hilbert kernel method, they are indeed conceptually quite different. The former have a bandwidth parameter that must be tuned to obtain statistical consistency and are in that sense similar to network-based approaches which also have parameters that need tuning. In stark contrast, the Hilbert scheme has no tunable parameter. The power-law exponent 3 of the wiNN-style schemes is also a tunable parameter satisfying 0 < 3 < d / 2, rather than the fixed value 3 = d for the Hilbert kernel. The price for the lack of a tunable parameter is that the Hilbert kernel estimator has slower convergence rates, discussed further below, but both have the same Bayes optimal performance in the large size limit. On the other hand, the lack of a tunable parameter is a fundamental benefit of the Hilbert kernel approach since it entails no parameter adjustments or training. This is greatly advantageous as the user does not have to worry about tuning bandwidth parameters or selecting models. We show with canonical, real- life examples from three different disciplines that the Hilbert kernel approach shows comparable performance with best in class ANNs for those problems.

[0053] It should be noted that data interpolation is a classical field with a large literature that has not had much interaction with machine learning until recently. In scattered data interpolation (see

[0030] for a review), the focus has been on the approximation error for non- noisy data (corresponding to the “bias” term in our analysis below). In contrast, the focus of the current paper is on the noise or the “variance” term. Interestingly, Hilbert-style kernels appear in the interpolation literature following original work by Shepard

[0047] ,

[0048] , for d= 2, for interpolating geospatial data. In the interpolation literature, the NW kernel is sometimes called the Shepard kernel, and arises naturally from the minimization of a locally weighted sum of squares (

[0030] , p.36) for function regression (LOESS)

[0049] , where the local fitting polynomial is a constant. However, there was no recognition or study in Shepard’s geospatial work or in subsequent data interpolation literature of the statistical consistency of the Hilbert kernel (and the necessity of the power law exponent being equal to the dimensionality of the input data). Devroye et. al. were the first to recognize and prove statistical consistency of this class of kernels, so we have adopted the nomenclature given by these authors based on the resemblance of the weights to the Hilbert transform.

[0054] In contrast with wiNN-style interpolation or Hilbert kernel interpolation, other interpolating regression methods, such as simplex interpolation

[0016] or ridgeless kernel regression

[0039] , are not statistically consistent in fixed finite dimension

[0036] ,

[0055] Given a dataset {xi,yt, i = 0,..., / / }, the Hilbert kernel estimator f (x) of the regression function x) (see Methods for problem setup and notation definitions) is a

[0056] Nadaraya-Watson style estimator employing a singular kernel,

[0057]

[0058] where wt(x) is the weight of the z-th data in the prediction f (x), for a given t / dimensional input x. Note that when x = Xi is an input vector from the dataset, wt(x = xi) = 1, wj(x = xi) = 0, for all j / = i, and the prediction exactly coincides with the dataset result, / (x = xi) = yt (true interpolation). FIG. 1 is an illustration of the Hilbert interpolation scheme in one dimension, a, an example is shown of the Hilbert kernel regression estimator in d= 1, both within and outside the input data domain. A total of 50 samples Xi were chosen uniformly distributed in the interval [0.25 0.75] andj', = sin(2^x,)+ / / , ith the noise ntchosen i.i.d. Gaussian distributed ~ A(0,0.1). The sample points are circled, and the function sin(2^x) is shown with a dashed line within the data domain. The solid line is the Hilbert kernel regression estimator. Note the interpolation behavior within the data domain and the extrapolation behavior outside the data domain, b, we plot the results of numerical simulations for the distribution p of the scaling I i / 1

[0059] variable7'’ ~ ru, withn~ >w(nhw)), and for n = 65536 (black line). This is compared to

[0060]

[0061] line), which has the predicted universal tail p(w) ~ w2for large w. c, numerical simulation of the expected value of the Lagrange function of the Hilbert kernel regression estimator in one dimension for a uniform distribution, as in FIG. la. n = 400 samples Xi were chosen uniformly distributed in the interval [0, 1] for 100 repeats, and the Lagrange function evaluated atxo = 0.5 was averaged across these 100 repeats (blue curve). The black curve shows the asymptotic form (1 + Z)]with Z = 2|x - xo| / TU.

[0062] We show in FIG. la an example of the Hilbert kernel regression estimator in one dimension, d= 1 (an application to classification by thresholding a soft classifier is shown in FIG. 10). Both the bias and the variance of the estimator can be visually seen, as well as the extrapolation behavior outside of the data domain. Note that in higher dimensions, the sharp peaks would have rounded tops, which increasingly flatten with dimension.

[0063] A generalized Hilbert kernel can be obtained by a change in coordinates, x — <b(x), where is an arbitrary diffeomorphism. This results in a large class of weighting schemes for the input data, each of which produces a statistically consistent estimate of the regression function with the same asymptotic convergence rates (see FIG. 16 for an example of the generalization using a coordinate transformation). Unlike ANNs or kernel regression, the change in coordinates does not necessitate a data-dependent parameter optimization step. Another generalization is obtained by replacing y, by a continuous function g(yt) in the estimator. This way, an estimate of / [ ()')|x] may be obtained, for example moments ofy may be estimated in this manner. These generalizations considerably broaden the original method.

[0064] Hilbert weights as in Eq.l, or other weights that are without adjustable parameters, permit the evaluation of data provenance from different sources of training data, pertinent to a new vector. Let xik denote the ith(i=l.. Mk) training data vector from the kthinput source (k=l.. Mk), with corresponding labels yik, where it is desired to attribute provenance to the different inout sources. Eq.1 can be used to compute the weights wik(x) for a new input vector x, for which a corresponding output y is generated using the procedure detailed above in Eq.1.2. Then, the relative weight of the predicted or generated output y (Eq.3) can be used to quantify the relative provenance or relative credit Prk (defined in Eq.4) corresponding to the output y and the training data attributable to the kth training data source:

[0065]

[0066] The quality or reliability of a prediction is an important consideration. For example, in scientific, biomedical, or other applications where input data sources may vary in quality or reliability, it is important to understand the quality or reliability of the output. For purposes of this discussion, we assume that associated with the ithtraining data pair from the kthsource (xi,kand yi,k), a numerical quality score Qk is available. For example, Qk could be a number between 0 and 1 assessing the degree of reliability of the kthsource, with 0 being completely unreliable and 1 being completely reliable. One could also equally well assign a quality score individually to each training input item, but without loss of generality, we will assign the same score to all the items from a given source. A natural question is, what is the quality or reliability of the predicted output?

[0067] If the prediction is computed using an ANN, the black-box nature of the network makes it difficult to assess the quality or reliability of the output since the provenance of the output in terms of the individual training data elements is not clear. However, given the method in Eq.3, 4 for determining provenance or relative credit attributable to a specific data item, it is possible to transparently estimate the quality / reliability Q of the output y corresponding to an input x, by computing the weighted sum of the quality measures of the individual sources using the provenance or credit Prk(x) scores for those sources, ie Q(%) = k=i Prk(x)Qk- A major advantage of the statistically consistent, parameter-free function estimation methodology above together with reliability assessment (Eqs.1-4) over the traditional ANN based approaches, is that altering the training data set (by adding or removing items) does not then entail a costly re-training of the network. Those training items can simply be added to, or left out of, the weight computation indicated in Eq.2. For example, if we have different training data sources of different quality, then one can re-evaluate the output for a new input on-the-fly by choosing a subset of training data elements of high quality passing some user-controlled threshold. Similarly, if data is being accumulated real time, these can be simply added to the training data store when computing the weights using Eq.2 without an expensive parameter optimization step.

[0068] Removal of items from the training data store, can be made necessary, for example, for privacy reasons. Suppose that the training data set was collected from individuals or groups, some of whom decide to opt out of the training data set. The corresponding records can be easily deleted from the training data store, before computing a prediction or output.

[0069] Thus, certain embodiments can weight the inputs in each output accurately (e.g., by assigning reliability coefficients to the input data, effectively grading the quality / trustworthiness of the input data), then we can (1) choose to include or exclude less reliable inputs to tune the reliability of the output and can (2) weight the reliability of the inputs to achieve a grading for the reliability / trustworthiness of the output. Clearly, in some circumstances, this such a reliability assessment may be irrelevant (for example, in a musicbased model, the inputs generally are all 100% reliable as they are all actual music / tunes, so the output should reliably be a tune). But in other cases, this is far more relevant. For example, if we have a data set where some of the information is more reliable such as in market data (given the issues around “black swan” as just one example) or news data (given conspiracy theories, opposing opinions, and “fake news” as another example), then it becomes much more critical to be able to rate the reliability of the inputs to judge the reliability of the output.

[0070] We proved that the excess risk of the generalized Hilbert estimator decays logarithmically with the sample size / / , under broad conditions and independently of the dimension c and that the risk is dominated by the variance term in a bias-variance decomposition, with the bias term being generically sub-leading (Theorems 4-5, i.e., no biasvariance tradeoff). We carried out a detailed characterization of the moments of the weight functions across sample realizations and conjecture that the full distribution has a scaling form showing power-law behavior. We also characterized the extrapolation behavior of the estimator outside the training data domain. The problem setup, precise theorem statements and proof outlines are presented in the Methods section, and detailed proofs are available in the Supplementary Materials in the attachments in the provisional patent application, reflected in FIGs. 10-16, for all theorems referred to here, under the corresponding theorem numbers.

[0071] We show (Theorem 8) under general conditions that the excess regression risk at the point x is asymptotically equivalent to the local variance c>2(x) reduced by the logarithm of the sample size (note: no unknown constants, no dimension dependence),

[0072]

[0073] We recognize that the dimension independence of the asymptotic form of the asymptotic regression risk is surprising, nevertheless we have rigorously established the stated theorems. Notably, subleading terms in the regression risk, and consequently the crossover to the asymptotic regime, do depend on data dimensionality. Our approach to exactly computing the asymptotic risk departs stylistically from risk bound proofs in the ML literature and is more analogous to exactly soluble models in theoretical physics where closed form calculations are possible. Although all n samples are used in the estimate, they contribute with quite different weights.

[0074] As discussed below, the effective number of degrees of freedom (d.o.f) contributing to the estimate can be shown to be ln( / / ). Thus, this formula may intuitively be interpreted as a reduction of the variance by the effective d.o.f.

[0075] For excess classification risk <5Ro / i(x) we show (Theorem 9) that for any e > 0, there exists a constant NX,E, such that for any n > Nx,s,

[0076]

[0077] Where the data distribution has nonzero density, we show (Theorem 1) that the ^-moments of the weights > 1 satisfy:

[0078]

[0079] In particular,1^- L o (■'')] ~ [»!»(«)]1implies E[S / W / 2(x)] ~ ln( / 7) so that the effective t / . ^. / 'defined by E[E, H’,2(X)]1~ ln(w). We also prove that the information entropy, S(x) = E[-S / W / (x)ln(w / (x))] ~ ln( / / _). We present heuristic arguments for a universal w2power-law behavior of the probability density of the weights in the large n limit (see FIG. lb for numerical evidence, details in the appendix).

[0080] The expected value of the weight function around a fixed data point is called the Lagrange function, which plays an important role in the classical literature on interpolation. For a point x within the support of the data distribution (assumed continuous), we are able to derive a limiting form of the Lagrange function (Theorem 2) by rescaling the Lagrange function with a local scale proportional to [y(x)wlog(w)]~1 / aIn this limit (denoted by limz), n +co, ||x - xoll +co (i.e., xo — x), such that IL / 7(x)||x - xo||dog(«) — Z, the Lagrange function

[0081] Zo(x) = EXvo[ii’o(x)] converges to a proper limit (see FIG. 1c for a numerical simulation, providing empirical support to the theoretical results),

[0082]

[0083] Asymptotic convergence rates are left unaltered with a smooth coordinate transformation x — <b(x). The asymptotic form of the Lagrange function for the transformed case may be obtained using the substitutions x — X= <b(x) and p(x) — R(X) =

[0084] / i(<b '(A))|6:X cA] (i.e., the density is transformed with the appropriate Jacobian). An optimal choice of <b, which governs subleading terms in the convergence, will depend on the details of the density p(x) and the function fix). However, the bias term depends inversely on p (see Eq. (53)), so a good heuristic for the coordinate transformation would be to make the density of X as uniform as possible over its support.

[0085] The lack of data-dimension dependence of the asymptotic convergence rates is remarkable, compared with other estimators where asymptotic risk worsens with increasing dimensions (curse of dimensionality). We expect the performance of the Hilbert estimator to improve with increasing dimensions compared with conventional estimators. Interpolating function estimators also extrapolate outside the support of the data distribution (see also the application to fish behavior modeling below). We characterize the extrapolation behavior of the Hilbert estimator, which can be seen in FIG. la, in theorem 10. Far outside the support, the estimate tends to the average value of the function over the data distribution.

[0086] A fundamental question in machine learning using parametric models such as ANNs is the model complexity, or the effective number of parameters in the model. Larger data sets may intuitively be expected to support more complex parametric models. Since the Hilbert scheme has no parameters, the usual notions of model complexity or parameter counting are irrelevant. However, one could ask how many independent sample points contributed effectively to the estimate at a given point x, corresponding to the notion of degrees of freedom in classical statistics. Thus, we define a heuristic, sample-dependent, and local estimate of an inverse number of degrees of freedom as,

[0087]

[0088] = iW2(x). If all the weights are the same and concentrated over N data, then wt = 1 / N for these data (and 0 otherwise), and we indeed find d.o.f. = N. In particular, when only one data point contributes to the prediction, d.o.f. = 1. Thus, the Hilbert scheme resembles an adaptive weighted nearest neighbor technique, with typically ln( / / ) nearby points contributing on average to the estimator, as defined by the inverse do.f.. Indeed, and in contrast with ANN-style methods, the actual number of data points contributing to an estimate can adapt to the provided input x.

[0089] Applications across three disciplines: Animal behavior, Control theory, and Probabilistic number theory

[0090] There is much current effort spanning several scientific and engineering disciplines to exploit ANNs as part of the toolset used in these disciplines to make predictions or model data. While the applications are domain specific, a broad architectural feature is the utilization of functions parameterized as ANNs and trained on a sample of input-output data pairs by optimization of an appropriate loss function. The ANNs have varied architectures across domains (CNNs, LSTMs, Transformers, etc.), with favored architectures evolving over time. Our proposal is to consider replacing these ANN function modules function learning with Hilbert kernel regression, particularly in applications where transparency and reproducible science are important, and large amounts of data are available, but the phenomena are complex enough that there are no strong priors for simple parametric models of specific forms, which can often be the case for modern applications.

[0091] We illustrate this approach in FIG. 2. FIG. 2 shows modular replacement of ANN parametric function module with Hilbert kernel module. Top: machine learning system schematic with a ANN-based or other parametric function module. Bottom: the same machine learning system schematic, but with the parametric / ANN function module replaced by the Hilbert kernel. Note that the parameter optimization or training module, where a loss function is minimized, is part of the ANN-based ML system on the top, but absent from the Hilbert kernel based system shown at the bottom. We show three possible ANN-based applications, where the overall system schematic may have other modules, but has an ANN function module which can be swapped for the Hilbert kernel module.

[0092] Note that in making this modular replacement, the learning or parameter optimization step based on a loss function is eliminated, with the tradeoff being that each function evaluation requires touching every training data point. It is important to note that while the systems into the Hilbert kernel function module will be inserted for function approximation could vary greatly with different data types and representations, the Hilbert module itself is universal and can indeed be implemented using the same underlying computer code. No handcrafting of the function or parameter tuning is required, which can be contrasted with the different and changing ANN architectures used in different application domains.

[0093] As can be seen in FIG. 2 the function learning module is generally embedded into a framework with other domain and problem-specific components, including data representations. In good scientific practice it is generally necessary to perform exploratory analysis of data to create this overall framework, which we are not trying to automate - we are separating a core component, namely function approximation from training data pairs, when such functions are too complex to model in the usual manner with a few parameters. The Hilbert approach provides the necessary function approximation without adding millions or billions of parameters to the overall scientific task, thus allowing for the clean study of specific hypotheses regarding the overall theoretical or modeling framework.

[0094] To illustrate the cross-disciplinary utility of the proposed approach, we demonstrate good performance of the modular swap of a trainable parametric model with the Hilbert kernel in three applications chosen to span three different disciplines: animal behavior, feedback control theory, and probabilistic number theory. These three applications further make use of a generative framework for process modeling which we develop below in the next section, but it should be clear from FIG. 2 that the modular replacement of a function box is broadly applicable across disciplines where parametric ANNs are being trained from input-output data pairs.

[0095] Interpolative generative modeling of time series

[0096] Interpolative regression, e.g., the Hilbert kernel estimator as defined above, applies directly to supervised learning and classification with labeled samples. Although stated for scalar or binary classification, the generalization to vectors (elementwise) is straightforward, and also to multi-class classification using a one-hot vector representation of the classes. Here, we propose applying interpolative regression to generative modeling, which is a major of area of current interest.

[0097] Generative Al architectures contain as core components ANNs that are subject to supervised learning using labeled examples (e.g., next token prediction in language models

[0050] , or predicting added noise in diffusion-based image generation models

[0051] ). These ANNs can be replaced by Hilbert kernel interpolators (other statistically consistent interpolating learners such as the wiNN algorithm

[0016] could be similarly used; in this paper we focus on the Hilbert kernel). This approach eliminates highly parameterized black boxes central to the generative architectures in favor of a parameter-free, pure-data approach, that brings the important benefits of transparency and ease of credit assignment. Here, we focus on generative or predictive modeling of time series, with applications to behavioral time series in animal behavior, feedback control of nonlinear dynamical systems, as well as series constructed from Zeta function zeros, as worked examples which we can examine in detail. Applying these ideas more broadly to generative Al is a rich area for future research.

[0098] Consider the generative modeling of time series (e.g., positions and velocities of fish, outputs of a driven dynamical system, tokens in language models, etc.). Parameterized ANNs (RCNNs, LSTMs, Transformers, etc.) have been previously used to predict the next time point given a past time window in an autoregressive manner. Instead, we propose Hilbert autoregression (note that Xi could be a scalar or vector)

[0099]

[0100] where / is a generalized Hilbert estimator of the prediction function from a lag window of length / which sets the dimension of the Hilbert estimator (more generally, a weighted average of such estimators weighted a power of T could be considered). Once training data is used to establish fn, the prediction model can then be used to generate samples of the time series xtfrom an initial segment of length T. Eq. (9) constitutes an autonomous dynamical system fully determined by the training data, and does not involve any model parameter fitting or learning, in contrast with existing approaches to autoregressive modeling. There is precedent to data-driven autoregressive prediction for nonlinear systems (e.g.,

[0052] ), but these approaches still have adjustable parameters and do not have theoretical guarantees of statistical consistency of the prediction function.

[0101] It should be noted that the generative process expressed by equation (9) can take as training input sequences corresponding to a database of music, word tokens corresponding to sentences from a natural language corpus, or other such sequences, to generate music, text, and so on. In such instances, it is of great interest to know the provenance of the generated music or text, namely the elements of training data that most influenced the generation of the output. Such provenance could, for example, be used to assign credit to the input sequences, to verify copyright issues associated with the generated music or language, or to allocate suitable compensation to each of the training data sources. Such attribution can be made possible by estimating provenance of the generated sequence using the procedure described in Eq.3, 4, combined with the generative model described in Eq.9. The proportional provenance or credit can be added up for all the generated sequence elements to construct a total provenance or credit corresponding to the individual data sources.

[0102] Since the ability to predict a signal also leads to an ability to predict a signal, Eq.9 using the Hilber kernel and its generalizations can also be used to drive a universal data compression algorithm for signals. To do this, the prediction has to be carried out in an online mode, where after each prediction, the true value of the observed signal, together with the past values, is entered into a live training data base which is continuously incremented.

[0103] Consider a signal which has been observed up to time Tobs. The past observations of the signal preceding the time Tobs can be used to generate a set of training data pairs, consisting of contiguous windows of time of length T to be the training input vector x, and the succeeding value at time T+l being the training output y. Then, using these input-output pairs as the training set, and the time window (xTobs-T+i, XTobs-T+2,..., XTobs ) as an input vector, a prediction XTpred may be generated for the time point T0bs+l using Eq.1-2. The residual Riobs+i = XTobs+i - XTpred will be small if the signal has predictability or equivalently compressibility. Thus, these residuals may be stored using fewer bits, therefore leading to data compression. An initial segment of the data, at least of length T+l, also needs to be stored together with the residuals. To ensure lossless compression, the original signal should be input as a sequence of integers, using appropriate preprocessing as needed. The predicted signal should be converted to the nearest integer using a fixed rounding procedure. In this case the residual will also be an integer and may be stored without loss or distortion.

[0104] Decompression reverses this procedure. During the decoding or decompression phase, the reconstructed signal up to a moving time horizon Tobs can be used as the training data set, similar to the compression phase, to make a prediction XTpred for the next time point Tobs+1. As for the encoding pass, the predicted value can be rounded to the nearest integer using the same procedure used during encoding. The stored residual Rrobs+i from the compressed data store can then be retrieved from the compressed data store and added to the prediction to obtain the decompressed value XTobs+i =XTPred+ Rrobs+i. This procedure can be iteratively repeated for subsequent time points until the entire signal has been reconstructed.

[0105] Examples of the generative procedure described in Eq.9 on simple stochastic processes including sinusoids and signals are shown in the Supplementary Materials in the attachments in the provisional patent application, reflected in FIGs. 10-16. We observed steady state, periodic (FIG. 15), and occasionally intermittent behaviors, as would be expected from dynamical systems theory. In addition, we observed “copying” behavior, where the dynamics copying parts of the training data after an initial transient (FIG. 14) or as part of periodic episodes (FIG. 15). Such copying behavior has been reported in Large Language Models, in the context of extraction of snippets of training data using appropriate prompts

[0029] and is of interest in the context of “stochastic parrot” style phenomena, i.e. the idea that the machine learning algorithm is essentially repeating the content of the data with some added noise due to limitations of the algorithm

[0053] , The extent of copying can be monitored using the entropy of the predictive weights as a function of time. In the real data example considered in the next section, where the training data consists of several time series fragments, we observe short episodes of near-copying behavior, but this does not dominate the generated time series.

[0106] The same procedure can also be used to predict one signal from another, or to predict the output of a driven, non-autonomous dynamical system if paired training data is available - e.g., for a pair of time series xt,yt, the predictive equation yt = fn(yt-\,yt-T.,..,yt-r t-i, t-2,..,xt-T), (10)

[0107] can be used, where / / / is constructed from a paired training set Xt,yt. During the prediction phase, only xtare observed, and the samples yt are generated recursively with a suitable initial condition. Modeling input-output relations or transfer functions is central to system identification

[0054] in control theory. Thus, we expect that the Hilbert framework will be applicable to the control of complex systems where system identification using parameterized models is challenging, and a direct data-driven approach provides an alternative to using black-box ANNs for the same purpose. We illustrate this with examples below to stabilize an inverted pendulum and controlling a driven chaotic map.

[0108] Hilbert generative model for realistic animal collective motion

[0109] Collective behaviors, and in particular, collective motions, are ubiquitous in nature, and are for instance observed in bacteria colonies, insects, bird flocks, or fish shoals [55- 57], The traditional approach to studying collective animal motion - e.g., collective swimming of fish schools - aims at producing data-driven mathematical models which exploit the inference of the social interactions between individuals [22-24, 58-62], Such approaches can lead to the direct implementation of the measured social interactions in analytical models, in good agreement with experiments [23, 24, 62], Deciphering the social interactions between individuals and their interactions with obstacles (e.g., the tank wall, in the fish context) offers in-depth understanding and direct interpretation of these social interactions and of the resulting collective behavior of fish, even when modifying the lighting of their environment [63, 64], As an application, the data driven mathematical model of

[0023] was recently implemented to command a robot fish interacting in closed-loop with 1 or 4 real fish

[0025] , a valuable tool to investigate the behavior of a fish group submitted to a controlled perturbation.

[0110] A more recent approach benefits from the progress in the applications of machine learning (ML). The social behavior of animal groups is directly learned by an ANN by using experimental collective trajectories as the training data [26, 65-68], In

[0026] , a deep learning model was designed, which quantitatively reproduces the long-term statistical properties of the motion of pairs of fish. This work

[0026] also emphasized the need to measure a set of stringent observables to properly assess the agreement between a generative model for animal trajectories and the short-term and long-term experimental dynamics. The deep learning model was shown to be in comparable agreement with experiments as the state-of-the-art mathematical model for the considered species

[0023] , Moreover, the deep learning model of

[0026] was also applied to another fish species, with similar success, and without retraining. This illustrates one of the main advantages of ML models, which can be reused without retraining to other species with similar motion patterns, contrary to mathematical models for which the social interactions must be specifically measured for each considered species

[0024] , On the other hand, ML models lack interpretability and do not provide detailed insight about the actual social interactions at play. In particular, and contrary to data-driven analytical models derived from the reconstruction procedure of [23, 24], ML models are unable to disentangle the contributions of the different interactions leading to the instantaneous behavior of an individual: attraction / repulsion and alignment with other individuals and avoidance of obstacles. Note that the ML model of

[0026] was also exploited to command a robot fish interacting in closed-loop with one real fish

[0068] ,

[0111] In the following, we show that the Hilbert interpolation scheme, exploiting experimental data for the dynamics of animal groups, constitutes a light and powerful generative model to produce realistic long-term trajectories of such groups. Hence, the Hilbert scheme provides a valuable alternative to other generative models for collective animal behavior, which we now explicitly describe in the context of the collective motion of small fish groups.

[0112] General implementation of the Hilbert scheme in the collective behavior context

[0113] We have implemented the Hilbert interpolation scheme as a generative model by exploiting the experimental trajectories for N = 2 and N = 5 rummy-nose tetra fish (Hemigrammus rhodostomus) using previously published data [25, 68], Adult rummynose tetra, of typical body length 3-3.5cm are social fish swimming in groups and can exhibit a highly coordinated behavior: highly polarized schools or vortex / milling schools.

[0114] We used 2.6 hours of data for N= 2 fish (18 experiments with different pairs of fish), and 9.4 hours of data for N= 5 fish (20 experiments), with all fish swimming in a circular arena of radius R = 25cm. Details about the data set are provided in the Methods section. We now detail the implementation of the Hilbert interpolation scheme in the present context, which is summarized in FIG. 3. FIG. 3 shows relevant fish variables and flow chart of the Hilbert interpolation scheme, a, Relevant variables for an individual: azimuthal angle, 0; heading angle, heading angle relative to the normal to the wall, (k. =

[0115]

[0116] - 0; distance to the wall, rw. b, Relevant variables for a pair of individuals: distance between the individuals, d relative heading angle between the 2 individuals, A<>; viewing angle at which the red focal individual perceives the other individual y. c, Snapshots from experimental videos from

[0025] , for N = 2 and N = 5 fish, d, Flowchart describing the implementation of the Hilbert interpolation scheme (for a memory M= 1) as a generative model for realistic fish trajectories.

[0117] We define the 2-dimensional positions, velocities, and accelerations of the TV Hilbert agents at a given time t as, respectively, Rt(Z), Vtf / )> AA Z), k =

[0118]

[0119] Similarly, the experimental positions, velocities, and accelerations of the N real fish for the / -th data point (occurring at time ti in some experimental trajectory) are denoted in lowercases: r*(h), v*(h), a^(h), k = The positions of the fish being measured with respect to the center of the tank at position (0,0), and the tank being circular and bounded, the theoretical mean of ri^f) and Ni^f) is zero, and we define the mean modulus of these quantities, or= (|| ||) and ov= (llvdl), also averaged over the fish k. We then define the rescaled and dimensionless positions and velocities for the fish and the Hilbert agents by ''rdtt) = r ti) / or, v' ti) = Nk ti) / ov, RX )=Kh / r, V / = ) / ^.

[0120] We now introduce the memory AT which will characterize the input vector x and the data vectors.vin the definition of the Hilbert weights, in Eqs. (1,2). Indeed, for each Hilbert agent or real fish, its configuration at time t is defined by its position at the current time, and its velocity at times t, t~ t,..., t-M t.

[0121]

[0122] In Eqs. (11,12), the input and data vectors involve rescaled positions and velocities so that all coordinates have the same typical magnitude of order unity. We note that these input and data vectors are of dimension,

[0123] d=N* 2 x [1 + (M+ 1)] = 2A( + 2), (13)

[0124] where the factor 2 comes from dealing with 2-dimensional position and velocity vectors, and the factor M+ 2 arises from taking into account the current position and M+ 1 present and past velocities of each agent / fish. Then, following the definition of the Hilbert scheme in Eqs. (1,2), and using Eqs. (11,12), the predicted accelerations, Ak(t) (k = 1,...,7V), for the Hilbert agents are obtained as a weighted average of the n experimental accelerations:

[0125]

[0126] (15)

[0127] A subtlety arises in matching the identity of the N Hilbert agents with the identity of the N real fish. A natural procedure consists in using the best matching permutation of the identities that have the closest match to a training data point (see the discussion in Methods).

[0128] Once the predicted accelerations, Ak(t) (k =

[0129]

[0130] of the Hilbert agents have been evaluated using Eqs. (14,15), their positions and velocities are updated according to the integration scheme,

[0131] Vk(t + At) = Vk(t) + Afc( At, ( 16)

[0132]

[0133] and the time is then updated, t t+ t. The procedure using Eqs. (11,12,14,15,16,17) and summarized in FIG. 3 is repeated to generate long trajectories of groups of A Hilbert agents. Finally, we have implemented a rejection procedure enforcing the agent to strictly remain in the arena (see the discussion in Methods). Below, we also evaluate the performance of the Hilbert scheme when this constraint is released.

[0134] Hilbert scheme for the collective dynamics of N=2 fish

[0135] For N = 2 fish, and following

[0023] , we quantify the individual behavior of the Hilbert agents or real fish by measuring the probability density functions (PDF) of (1) their speed, (2) their distance to the wall, rw, and (3) their heading angle relative to the normal to the wall, 0W(see FIG. 3a for a graphical representation of these variables). The collective behavior is quantified by the PDF of (4) the distance d between the pair of individuals, (5) the group polarization,P~

[0136]

[0137] which reduces to^:::|«»(A<? / 2^ for N= 2, where A<) is the relative heading angle, and (6) the viewing angle at which an individual perceives the other individual, y / (see FIG. 3b for a graphical representation of these variables). Finally, we quantify temporal correlations by (7) the mean squared displacement, Cxt) = (( (t + t) - n( / j)2), (8) the velocity autocorrelation, (7(1) = ( t + t)

[0138]

[0139] and (9) the polarization autocorrelation, Cp(f) = (P(t + / ) • P(f)), where the average is over the reference time / , and over the individual k (only for Cx(i) and C( / )). Time correlation functions are notably difficult to reproduce and represent a stringent test of a model

[0026] , The comparison of these 9 observables for the experiment and the Hilbert interpolation scheme for M = 2 is presented in FIG. 4. FIG. 4 shows a comparison of the behavior of real and Hilbert fish for N = 2 individuals. The different panels show the 9 observables used to characterize the individual (a-c) and collective (d-f) behavior, and the time correlations in the system (g-i): a, PDF of the speed, V b, PDF of the distance to the wall, rw; c, PDF of the heading angle relative to the normal to the wall, 0W; d, PDF of the distance between the pair of individuals, d e, PDF of the group polarization,7' ~ d0S(A< V2)( where A<) is the relative heading angle; f, PDF of the viewing angle at which an individual perceives the other individual,

[0140]

[0141] See FIG. 3a and b for a visual representation of these different variables, g, Mean squared displacement, Cx t and its asymptotic limit, Cd00) = 2(r2) ~ 900cm2(dotted lines); h, Velocity autocorrelation, C( / ); i, Polarization autocorrelation, Cp(f). The black lines correspond to experiments, while the red lines correspond to the predictions of the Hilbert generative model for a memory M = 2 and for 4 hours of simulated trajectories. Vertical dashed lines correspond to the mean of the corresponding PDF (see also Extended Data Table 1). We also report the mean and the standard errors for each PDF in Extended Data Table 1. We find an excellent overall agreement between the Hilbert and experimental results, with a significantly better accuracy than the reported results for the data-driven mathematical model

[0023] or for the machine learning approach [26, 68], Both real fish and Hilbert agents move at a typical speed of 11,5cm / s. They also remain close to the wall (mean distance rwof 4cm). This is a consequence of the burst-and-coast swimming mode of rummy-nose tetra [23, 63], consisting in a succession of short acceleration periods (kicks / bursts of typical duration 0.1s), each followed by a longer gliding period (typically 0.5s) in a near straight line, ultimately preventing the fish to effectively escape the concavity of the tank wall. This is confirmed by the sharp peaks near - but slightly below

[0023] - ±90° in the PDF of the heading relative to the normal to the wall, 6L, showing that fish and Hilbert agents are often almost parallel to the wall. Finally, the motion of the pairs of fish and Hilbert agents is highly coordinated: both individuals remain at a typical distance of 8cm and the pairs are highly polarized (P = 0.95). Finally, the PDF of the viewing angle y at which an individual perceives the other quantifies the relative placement of the 2 individuals. The wide peaks near yt = 0° and y / = ±180° reveal that an individual tends to stay in front or behind the other one.

[0142] Study of the temporal correlations clearly shows the superiority of the Hilbert generated trajectories in mimicking the real behaviors over both the analytical and ML models. The mean squared displacement, C^f), starts with a quadratic (ballistic) regime, Cx(f) ~ (v2) / 2, followed by a short linear (diffusive) regime, Cx(f) ~ Dt, interrupted due to the bounded tank. Finally, C.( / ) exhibits rapidly damped oscillations before leveling at Cv(co) = 2(r2), when the two positions at times t + t and (become uncorrelated. The Hilbert model reproduces quantitatively the ballistic and diffusive regimes, and the main oscillation, as well as the asymptotic level. The velocity autocorrelation, Cv(f), decays with damped oscillations from its maximal value,

[0143] Cv( / = 0) = (v2).

[0144] Note that the ML model of

[0026] and, to a lesser extent, the behavioral mathematical model of

[0023] , fail at reproducing the experimental Cx(f) and Cv(f).

[0145] Finally, the model also reproduces the initial decay of the polarization autocorrelation, Cp(f), from Cp(t = 0) = (2) ~ 0.94, and its main oscillation. We illustrate these results in SI Movie SI, which shows a 2-minute collective dynamics of 2 Hilbert agents. We have also studied the impact of the memory M. We find that using no memory (M= 0) for the Hilbert input and data vectors of Eqs. (11,12) would not produce coordinated trajectories. Our analysis below of the dynamics without the rejection procedure enforcing the presence of the wall will confirm that a memory M= 0 fails to reproduce the correct fish behavior. We found that M= 1, M= 2, and M= 3 lead to comparable results, while the agreement with experiments is degrading for a memory in the range M= 4-10. A possible reason could be the increase of the dimension d of the input and data vectors with AT (see Eq. (13)), resulting in a scarcity of data in the tZ-dimensional configuration space (d= 16 for M= 2; d= 48 for M = 10).

[0146] In addition, we have tested the stability of the Hilbert scheme when not implementing the rejection procedure strictly enforcing the presence of the tank wall (see Methods). For M = 0 (no memory), the 2 agents quickly escape the tank, become independent, and wander several meters away from the tank, confirming that the Hilbert scheme for M= 0 is unable to capture the social interactions between fish nor the presence of the wall. We ascribe the failure of the M= 0 Hilbert model to the fact that the input and data vectors x and Xi in Eqs. (11,12) do not encode enough information to evaluate the similarity of a fish configuration and a Hilbert agent’s configuration. In fact, when M > 0, these vectors now implicitly encode an estimate of the acceleration, through the knowledge of the velocity at time / , but also at past times t - - M t. Indeed, for M > 0, we find that the Hilbert agents remain cohesive, stay inside the tank for the vast majority of the simulations (87% for M= 2), and only make brief and small (1,3cm away from the tank wall, on average) excursions outside the tank. This is illustrated for M = 2 in the FIG. 11, which corresponds to FIG. 4 but without the rejection procedure. The deep learning models of [26, 68] present a similar stability without the rejection procedure, showing that both Hilbert (for M > 0) and ML models can implicitly capture and learn the presence of the wall from the data. In fact, note that the deep learning models of [26, 68] also exploit a memory of the past, encoded in the input data (with an effective M= 5) and in their architecture involving LSTM layers. SI Movie S2 shows a 2-minute collective dynamics of 2 Hilbert agents without enforcing the presence of the wall, confirming that the Hilbert interpolation scheme has implicitly learned the presence of the wall from the data.

[0147] A valuable advantage of the Hilbert interpolation scheme compared to ML methods is that it explicitly gives the weight wt(x) of a given data x, involved in the current prediction, for a given input x (see also the final section about credit assignment). To quantify the effective number of data, N, involved in this prediction, we define the entropy,

[0148]

[0149] where the weights Wi(x(f)) are defined in Eq. (15). If the weights at a given time t are equi distributed over N data, one obtains S = -Nx1 / N Iog2(l / N) = log2(N), and 2S= N indeed represents the number of contributing data. In particular, in the presence of “copying” behavior, N = 1 and S = 0. As shown below, these theoretical results following from our analytical considerations are consistent with the experimental observations. In FIG. 12, we show a 2-minute time series for S(f) exhibiting very short periods when S ~ 0 (N ~ 1), meaning that the Hilbert prediction is essentially using / copying a single experimental data / configuration to compute the predicted accelerations. On the other hand, this short time series also presents 3 periods when S > 6 (N > 64). In the FIG. 12, we plot the PDF of S(f), p(S), which presents an integrable divergence near 5= 0 p(S) ~ S-1 / 2), and a Gaussian tail for large S. Over 3 effective hours of simulations, we recorded entropies as large as S ~ 15, corresponding to an effective number of data involved in the prediction of order N ~ 215~ 32768. Compared to ANN methods, the Hilbert interpolation scheme is able to adapt the effective number of data points used for the prediction to the properties of the input vector.

[0150] It is important to note that the close match between the observables and correlations from the Hilbert generated trajectories and the experimental observations cannot simply be attributed to copying behavior. A study of the weight entropy (FIG. 12) shows only brief episodes of near-copying behaviors, shorter than the timescales involved in the correlation functions and too infrequent to explain the results.

[0151] Hilbert scheme for the collective dynamics of N=5 fish

[0152] For N= 5 fish, we quantify the individual behavior and the temporal correlations of the Hilbert agents or real fish with the same 6 observables as for N= 2, but we adapt 2 of the 3 collective observables to the larger fish group [62, 63]: PDF of (4 ) the distance d between an individual and its nearest neighbor, and (6)’ the radius of gyration of the group, Royr =q-i) ||R- R / ll, which is a measure of the size of the group. FIG. 5 shows a comparison of the behavior of real and Hilbert fish for N = 5 individuals. The different panels show the 9 observables used to characterize the individual (a-c) and collective (d-f) behavior, and the time correlations in the system (g-i): a, PDF of the speed, V b, PDF of the distance to the wall, rw; c, PDF of the heading angle relative to the normal to the wall, 0W; d, PDF of the distance between an individual and its closest neighbor, d e, PDF of the group polarization,; f, PDF of the radius of gyration of the group, Jtayr. See FIG. 3a and b for a visual representation of the main variables, g, Mean squared displacement, Cx(f), and its asymptotic limit, Cx(co) = 2(r2) ~ 680cm2(dotted lines); h, Velocity autocorrelation, C( / ); i, Polarization autocorrelation, Cp(f). The black lines correspond to experiments, while the red lines correspond to the predictions of the Hilbert generative model for a memory M = 2 and for 2 hours of simulated trajectories. Vertical dashed lines correspond to the mean of the corresponding PDF (see also Extended Data Table 1). The horizontal and vertical scales of each graph are the same as that of the corresponding graph in FIG. 4, for N= 2.

[0153] The comparison of the 9 observables for the experiment and the Hilbert interpolation scheme for M = 2 is presented in FIG. 5, where a fair general agreement is obtained, comparable to that of the data-driven mathematical model [62, 63], We also report the mean and the standard errors of these different observables in Extended Data Table 1. The PDF of the speed and distance to the wall are in good agreement with experiments and show that groups of 5 fish and Hilbert agents are moving significantly faster than pairs (mean speed 15.3cm / s for N= 5 vs. 11.5cm / s for N= 2), and significantly farther from the wall than pairs (mean distance to the wall 7cm for N= 5 vs. 4cm for N= 2). The PDF of the angle relative to the wall is less peaked than for N= 2, showing that 5 fish or Hilbert agents are less often parallel to the wall than corresponding pairs. However, for N= 5, the Hilbert agents are slightly less coordinated than real fish, with a larger distance between nearest neighbors (4cm for fish vs. 5cm for Hilbert agents). Note that in both cases, individuals are, on average, closer to each other than for N= 2 (mean distance of 8cm) [62, 63], Accordingly, the typical size of the group (quantified by the radius of gyration) is significantly smaller for fish than for Hilbert agents (7.6cm vs. 10.4cm), which represents the worst agreement observed so far. Yet, groups of N = 5 fish and Hilbert agents are extremely and almost equally polarized (P = 0.96 for fish vs. P = 0.94 for Hilbert agents). Still, considering that the trajectories of the 5 Hilbert agents are integrated independently, it is remarkable that the Hilbert model produces such cohesive and highly polarized groups. The Hilbert model reproduces fairly well the 3 experimental correlation functions, albeit with a longer persistence of the oscillations than for fish. The mean squared displacement Cx(f) presents a smaller amplitude than for A= 2 (see FIG. 4), since fish and Hilbert agents are, on average, moving farther from the wall tank for TV = 5. The period of the oscillations is shorter for N = 5, since fish and Hilbert agents move faster than for N = 2, and these oscillations are better marked than for N = 2. The velocity autocorrelation, Cv(f), and the polarization autocorrelation, C ( ), also display larger, faster, and more persistent oscillations than for N= 2. These results indicate that 5 fish or Hilbert agents maintain strong temporal correlations for larger traveled distance than corresponding pairs. We illustrate these results in SI Movie S3, which shows a 1-minute collective dynamics of 5 Hilbert agents.

[0154] In this section, we have applied the generative Hilbert scheme to the modeling of animal collective motion. The statistical properties of the generated trajectories closely match experimental observations as quantified by 9 observables probing the instantaneous individual and collective behavior, as well as the temporal correlations. These observables offer a stringent test of the model, and in the present fish context, their analysis indicates that the Hilbert scheme provides a new and complementary approach to data-driven analytical modeling and ANN-based modeling, with equally good or better predictive capabilities. The analytical fish models of [22, 23] follow a canonical tradition in scientific theorizing, using simple and interpretable low-order (continuous or discrete) ordinary differential equations. On the other hand, both the ANN-based and Hilbert generative models are purely data-driven and neither offer a similar degree of interpretability as mathematical models. However, we posit that the Hilbert scheme nevertheless offers significant advantages over the ANN-based model.

[0155] The Hilbert generative model leads to a straightforward implementation, consisting of a few lines of code (see FIG. 3). There is no training, parameter tuning, or model selection, and it offers strict reproducibility, portability, and transparency, which is particularly important for scientific applications. This framework also permits the reuse of generated data in real time to augment the experimental dataset. Moreover, the relative contribution of training data points contributing to a specific prediction is transparently characterized using the corresponding weights, and the procedure flexibly adapts those weights across training data points for any given prediction. In conclusion, the present work shows that the Hilbert scheme constitutes an advantageous alternative to ANN-based generative approaches in the context of the very active field of collective animal motion and behavior.

[0156] Hilbert Predictive Control of nonlinear dynamical systems

[0157] Control systems are central to modem technology. The goal of control engineering is to ensure that systems have behaviors suited to the task at hand. Active control of a system generally involves two steps: modeling the response of the system state to controlled external inputs (system identification), and using this model to design appropriate inputs to make the system state follow a desired dynamical trajectory (controller design). In complex modern applications, the system models are increasingly data-driven, and are often exploiting ANNs. As with other applications of black box networks, the precise role of the network-based system models remains unclear.

[0158] In keeping with the central idea in this work, we show that given a control objective, neither parametric system models not controllers are required for the control of dynamical systems. Strikingly, dynamical control can be achieved by simply interpolating previous observations of system responses to inputs, and directly determining the control action using a stochastic optimization of the future cost over multiple scenarios. Our approach is similar in spirit to the behavioral approach to control due to Willems

[0069] , However, this earlier framework exploits the linear dependency between sequences of lag vectors of the system outputs and inputs, which only exist for linear dynamical systems. In contrast, our approach to system modeling based on Eq. (10) is applicable more generally nonlinear systems including discrete time nonlinear maps without making a local linear assumption.

[0159] We use training data consisting of past input-output observations to predict the future input-output behavior of the system using the Hilbert-based transfer function predictor Eq. (10). Our work is analogous to the framework of Model Predictive Control (MPC) where the system model is replaced by Eq. (10) and where the controller is replaced by direct stochastic optimization of the control cost. It is important to note that the MPC framework does not automatically provide a control function, but is an iterative procedure that utilizes both a plant model as well as a control function or controller.

[0160] In MPC, the objective or cost function based on the system model and a controller, both of which are usually tuned to the application, is optimized over a future horizon to find the optimal control input over that horizon. Subsequently only the control input for the first time step is implemented, following which the entire procedure is iterated. Since we do not have either a parametric plant or controller model, we cannot perform such an optimization. Instead we directly optimize the control objective using a stochastic gradient descent procedure where only cost function evaluations are utilized (see Methods). It is important to note that there is no controller involved in the usual sense, namely a known function that takes system state as input and produces a control action as output. Instead, the control action is directly obtained by stochastically optimizing a finite horizon cost. The closest to this is the Behavioral Control framework applied to an MPC context, but there a linear system is assumed, and a deterministic controller is derived from the data Hankel matrices by solving an optimal control problem at each time step. In the ANN case, ANNs are used for either or both the system model and the controller.

[0161] We apply our approach to two disparate nonlinear control problems: stabilizing an inverted pendulum around the unstable vertical equilibrium (FIG. 7a), and stabilizing a driven logistic map (FIG. 7b) defined by

[0162] xt+i = (1 - K) x Axt(l - xt) + K x ut (19)

[0163] also around an unstable fixed point.

[0164] FIG. 6 shows Input-output prediction using Hilbert autoregression (Eq. (10)). FIG. la shows offline and online prediction of an input driven, linear two-state system, as well as online prediction with a double integrator where the acceleration is predicted and integrated twice to obtain the time series prediction. The variance explained by the offline and online autoregressive prediction methods are 66.18% and 94.20%. When the acceleration is predicted, and integrated twice to obtain the signal prediction, 99.87% of the variance is explained. FIG lb shows similar predictions for a driven inverted pendulum, where Q = n corresponds to the pendulum positioned vertically down. The variance explained by the offline and online autoregressive prediction methods are 65.46% and 87.10%, and for the online double integration based prediction, 99.87% of the variance is explained. Note that the variance explained is similar for the linear and the nonlinear dynamical system examples. FIG. 1c shows predictions of a driven logistic map (Eq. (19) with A = 3.99, Xi = 0.25) using offline and online autoregressive prediction. The prediction is fairly accurate even for this highly nonlinear map: the offline predictor has a residual of 5.46% (median absolute deviation) and the online predictor has a residual of 1.54%. The top panel of 1c shows one representative instance, whereas the normalized median absolute deviations below are computed from ten independent runs with different random seeds. Notably, the offline prediction starts with a similar error as the online prediction, but over time the offline prediction deteriorates somewhat.

[0165] In FIG. 6, the effectiveness of the Hilbert-based transfer function is shown for three increasingly nonlinear cases: a linear two-state system whose impulse response is a damped oscillation (FIG. 6a), a torque-driven pendulum (FIG. 6b) and a driven logistic map (FIG. 6c). In each case, the top row exhibits the output of the autoregressive predictor (Eq. (10)) in blue superposed on the true signal in black (using lag windows of size 5, 10 and 3 respectively). This case is labeled Hilbert offline, since the prediction is carried out using a fixed past episode of training data. The red curves correspond to the same procedure applied in an online manner (see Methods), in which at each time step, a measurement of the true system output is added back to the training data together with the control input applied. It can be seen that the online predictor performs better than the offline predictor. For the smooth dynamical systems (FIG. 6a and 6b), the result is even better if the acceleration is predicted (see Methods), rather than the signal directly, and the acceleration integrated twice to obtain the prediction in an online manner. In this case, the error as quantified by the unexplained variance is ~ 0.1% and the black curve (original) and prediction (red symbols) are visually indistinguishable. For the driven logistic map case, the dynamics is non-smooth and predicting the acceleration does not improve performance, but the online predictor has a residual of only 1.54%. The offline prediction slightly deteriorates over time but still has a residual variance of 5.46%. It is important to re-emphasize that in each case, precisely the same (short) code together with the training data is used for the predictive step: there is no system modeling involved.

[0166] Armed with the input-output predictors, we can proceed to the next step of controlling the outputs to a desired target. To achieve this goal, we follow the basic idea behind the MPC framework, which involves an iterative process in which at each time step, a prediction is made over some time horizon T for command inputs which minimize a suitable cost (we chose constant targets and a quadratic cost on the output signal). In the existing MPC framework, a parametric system model trained on previous data is used to make the prediction. In our case, the parametric model is replaced by the Hilbert-based predictor. Not having a parametric system model and not using a linear systems assumption means that usual controller design methods cannot be used. We developed a stochastic gradient descent optimization approach that requires only function evaluations, in which a directional gradient is estimated in a randomly chosen direction

[0070] (see Methods) and an appropriately scaled step is taken in that direction. This approach works for stabilizing two very different nonlinear dynamical systems: an inverted pendulum and a driven logistic map (FIG. 7). The proposed procedure for data-driven control is new to the best of our knowledge, and we termed it Hilbert Predictive Control (HPC).

[0167] FIG. 7 shows Hilbert Predictive Control of nonlinear dynamical systems. The left panels show that an inverted pendulum can be controlled to its unstable equilibrium with the MPC approach, where the driven system dynamics is predicted using Hilbert interpolation illustrated in FIG. 6 rather than a parametric model. For an initial period, the pendulum is driven by Gaussian white noise input to generate training data. The controller is activated at the black vertical line, following which the pendulum is driven to its unstable equilibrium position, and maintained there with small fluctuations (top left). As a comparison, the true pendulum equations used to generate the dynamics are used in the same MPC framework to show a baseline controller (bottom left). The plots show arccos(co (0)) with the angle Q = n corresponding to the vertically up configuration, for ten independent runs with different random number seeds. The RMS error achieved in steady state for the Hilbert MPC approach is 0.0129 radians, compared to a baseline RMS error using the true system model of 0.0027 radians. Similarly, the top right panel shows the Hilbert predictive controller approach applied to the control of a driven logistic map Eq. (19) around an unstable fixed point of the driven system for a constant input (xtarget= 0.72, utarget= 0.4668). Uniformly distributed white noise is input during the training period and the controller is turned on at the black line. The controller is effective in stabilizing the system around the unstable fixed point (the control is turned on at the black vertical line). The control is statistically indistinguishable from a baseline case where the underlying driven logistic map is used instead of the Hilbert prediction. We quantify this using the standard deviation of x - xtargetnormalized by the corresponding standard deviation when the controller is not turned on. This ratio, which should ideally be zero, is 0.1434 ± 0.0063 (sd) for the Hilbert case and 0.1479 ± 0.0078 (sd) for the baseline case, where the standard deviations are computed with ten independent runs with different random number seeds.

[0168] FIG. 7 shows that HPC can stabilize nonlinear dynamical systems around unstable fixed points. In FIG. 7a a pendulum is controlled to its unstable equilibrium, pointing upwards. The pendulum is initially subjected to a Gaussian white noise input to generate training data. The controller is then turned on at a time marked by the black line, causing the pendulum to swing up and remain near the unstable equilibrium point. The top panel in FIG.

[0169] 7a shows 10 independent runs with different random number seeds, with an RMS (root mean square) departure from target of 0.0129 radians. The bottom panel in FIG. 7a shows ten independent runs with the same seeds, where the true pendulum model is used in place of the Hilbert predictor, retaining the same derivative-free optimization scheme, giving an RMS departure from target of 0.0027 radians. In FIG. 7b the top panel shows the result of applying HPC to the driven logistic map Eq. (19). After an initial period of uniformly distributed white noise input (confined to [0,1]) to generate training data, the control is turned on, and the output quickly collapses to the target under the HPC approach, where the system model is replaced by the Hilbert predictor Eq. (10). In this case, we find that if the true system model is used instead of the Hilbert predictor (keeping the controller design optimization method the same), the results are indistinguishable. The cumulative distribution function of the deviation from the target position of the Hilbert controller (FIG. 7b, bottom panel red curve; normalized standard deviation 0.1434 ± 0.0063) is statistically indistinguishable from the corresponding deviations after applying the MPC controller using the true system model (FIG. 7b, bottom panel blue curve, normalized standard deviation 0.1479 ± 0.0078).

[0170] In conclusion, it is possible to control nonlinear dynamical systems around unstable fixed points using HPC. Notably, the control algorithm for the two widely divergent cases is identical and does not involve parametric models or controllers and is implemented by the same few lines of code. Note that there is however a cost of Monte Carlo optimization requiring the stochastic testing of many control inputs at each time step, which is the price to be paid for having no controller.

[0171] To re-emphasize, while it might not be difficult to find ANNs to control either the inverted pendulum or possibly the driven logistic map, the crucial point is that both of these very different systems are controlled by a single, universal parameter free algorithm implemented using the same few lines of code involving stochastic optimization but without parameters to tune. We do not propose that this approach will be able to solve all complex control problems out of the box, as control engineering problems are sensitive to many practical details including the limitations of the sensors and actuators involved. Bread-and-butter controls problems in typical engineering settings are often adequately addressed by simple approaches such as tuning univariate PID controllers at fast time scales, coupled with high level decision making at slower time scales. However, to the best of our knowledge the HPC approach is conceptually new and and opens up an interesting direction of research, both in theoretical terms, and in applications where the plant dynamics are complex and non- stationary, and the effect of controller actions have to be learned from data.

[0172] In addition to controlling a dynamical system through feedback, the approach developed above may also be used with suitable modification to carry out reinforcement learning. In the control system application, although a model of the dynamical system or of the control action is not generally known, a control cost function is known or assumed. Such a cost function is optimized in order to choose a control action. What happens if the cost is not known, but after taking each control step, a value of the cost (or equivalently, a reward) is revealed or can be measured? This is the situation with reinforcement learning. Assuming that the reward (or equivalently, negative cost), obtained at the next time step t+1 is given by rt+i, and it is desired to optimize this reward or the accumulated reward over some time horizon. If the reward at the next time step rt+i is to some extent predictable given the current state xt, the current reward rrand the control action ut, and if one has access to a prediction of the reward rPred.t+i then one can substitute the control cost in the above procedure with the predicted reward and carry out the same stochastic optimization of potential future actions.

[0173] Note that this reward prediction may be carried out using the Hilbert procedure for transfer function estimation described in Eq.10, where the input series is chosen to be the past series of states xt, actions ut and rewards rt, and the Hilbert predictors for the state as well as the reward at the next time step are computed, ie xt= xH(xt-i,xt-2,..,yt-T,' rt-\,rt-2,..,rt-T,' ut-i,ut-2,..,iii-T) and i =frH(xt-i,xt-2,..,yt-T rt-\,rt-2,..,rt-T,' ut-i,ut-2,..,ui-r) in an analogous manner to Eq.10.

[0174] Hilbert predictor for the zeros of the Riemann Zeta function

[0175] The Riemann zeta function, defined by A ~

[0176]

[0177] ! l >l; and by its analytical continuation elsewhere), plays a significant role in number theory due to its close relation with the distribution of prime numbers

[0071] , Central to this connection is the Riemann hypothesis, which conjectures that all nontrivial zeros of < are of the form

[0178]

[0179] and are hence all located on a line in the complex plane. If confirmed, this conjecture would provide validation for a plethora of theorems in number theory that have been proven under the assumption of the Riemann hypothesis. Even more intriguing is the stronger conjecture that the imaginary part of the zeroes, E (that we will assimilate to the zeroes of, are the eigenvalues (i.e., energy levels) of a quantum Hamiltonian and are hence aligned on the real line

[0072] .

[0180] This last conjecture is numerically confirmed by the fact that the high-energy levels of non-integrable quantum Hamiltonians and large “Riemann energies” E share the same statistical properties as the eigenvalues of large Hermitian random matrices with Gaussian entries

[0073] , In particular, the spacings between successive eigenvalues (for quantum or random Gaussian operators) or Riemann energies, normalized by the local mean spacing, obey a universal statistics. More precisely, in the context of Riemann energies, we define St = ( + - Ei) / EE, as the difference between the (z + 1 )-th and / -th zero of normalized by the

[0181] mean local spacingyJ g(U) i- -l]. Then, for large z, the variable St effectively behaves like a true random variable following the universal GUE statistics given by random matrix theory, with a PDF very well approximated by the Wigner surmise,

[0182]

[0183] Random matrix theory also makes precise predictions for the correlation function between different energy levels, which are numerically verified for the zeros of the Riemann function. In order to avoid the need to normalize the spacings by the local mean spacing, it is customary to introduce the ratio between two successive spacings, = (E+i - Ei) / (Ei - Ei-i), which now behaves like a quasi stationary random variable. Again, the PDF of the ratio r for Riemann zeros is in excellent agreement with the prediction of random matrix theory

[0074] ,

[0184] By assimilating the index z of a zero to a time, we have exploited the Hilbert scheme to predict the next ratio = r(f), knowing x = (r(t - 1), r(t - 2),...,r(t - )), the latter d-dimensional vector constituting the input for the Hilbert kernel in Eq. (9). As discussed in the Methods section, since values of the ratio r span several orders of magnitude, we chose to deal with the log-ratio log(r) instead of r. In parallel, we have trained a Transformer

[0050] to achieve the same goal (see details in Methods). For the training data, we have used 100000 (log-)ratios starting at the 1000000th ratio (to lie deep in the “high-energy” universal regime; table of zeros retrieved from

[0075] ). Obviously, the Hilbert and Transformer predictors cannot truly predict the actual next term of a sequence of ratios, since the sequence of zeroes behaves effectively like a true random variable. However, they can both exploit the strong correlations between zeroes to offer a better estimate than, say, the average ratio (r), while not beating the Bayes limit. For the Hilbert predictor, the results naturally depend on the length d of the input sequence, d = 1 and d = 2 lead to a mediocre performance, while the performance saturates rapidly with d, presumably due to the limited range of the correlations between Riemann zeros or ratios. In the following, we discuss our results for d= 5.

[0185] FIG. 8 shows predicting zeros of the Riemann function, a, PDF of log(r / rpred) (similar to a signed relative error), where r is the actual observed ratio, and rpredis the prediction of the Hilbert (d= 5; red line) and Transformer (blue line) predictors (200000 predictions). For reference, we also plot the corresponding PDF for two other natural predictors: / 'Pred = ( / ') (the expected value of the ratio; light blue dots) and rpred = r (a ratio randomly drawn in the dataset; black dots). The vertical dashed lines are the corresponding mean error, (|log(r / rPred)|). b, PDF of the normalized absolute error (r - rPred) / {r), and using the same notations. In both graphs, the PDF for the Hilbert and Transformer predictors are nearly identical and result in similar errors, significantly lower than for the two other simple predictors. Panel c shows a sequence of 50 successive predictions of the Hilbert (red dots) and Transformer (blue dots) predictors, along with the actual observed values of log( / j (black dots), d shows the same 50 predictions as a function of the actual observed log-ratio, along with the linear correlation measured between the 200000 predictions and observed values for the log-ratio (black dashed line), corresponding to a Pearson correlation coefficient R = 0.7.

[0186] In FIG. 8, we show that, although the individual predictions of the Hilbert and Transformer predictors are different, they share the same statistical properties and a strong correlation with the observed ratio. Both predictors produce a much smaller mean error (and a narrower PDF of the error) than when using the mean ratio ( / ') or a ratio randomly drawn in the dataset as a simple predictor for the next ratio.

[0187] Moreover, the fact that the PDF of the relative and absolute error for the Hilbert and Transformer predictors coincide suggests that both have reached the Bayes limit. In addition, the Pearson normalized correlation coefficient between 200000 observed log-ratios (outside the training range) and the corresponding Hilbert and Transformer predictions is found to be R = Q.rl for both predictors.

[0188] Finally, in Fig. 9, in order to visualize the correlations between ratios captured by both predictors, we plot the mean value of the prediction for the next log-ratio, (log(r( / +l))), obtained for all observed sequences with a given r(f) and r(Z-l). Again, the results for the Hilbert and Transformer predictors are very similar and are in good agreement with the actual mean value of the next observed log-ratio. FIG. 9 shows correlations of the zeros of the Riemann function. For the Hilbert (d= 5; red surface in left panel) and Transformer (blue surface in right panel) predictors, we plot the mean value of the prediction for the next log- ratio, (log(r(t + 1))), obtained for all observed sequences with a given r(t) and r(t - 1) (up to the size of the binning box). We have only retained bins containing at least 20 data when exploiting the 200000 predictions whose PDF is shown in FIG. 8. The gray surface corresponds to the actual mean of the observed next log-ratio, and is almost indistinguishable from the Hilbert and Transformer predictions. We provide a python code (see Data availability) which allows to reproduce and interact with this figure to change the angle of view (synchronized between both panels).

[0189] In conclusion, the Hilbert and Transformer predictors applied to the zeros of the Riemann function exhibit a very similar performance, and have presumably both reached the Bayes limit.

[0190] Credit assignment in interpolative generative modeling

[0191] An interpolating regression function is by construction linear in the labels, with the output being a weighted sum of training data labels. Furthermore, in the Hilbert kernel, the weights do not have any adjustable parameters. This provides a transparent route to credit assignment among different training data fragments. If an autoregressive generative model is used to create a time series xt(e.g., a musical wave form), then for any given time point t, the contribution of each training data point i is given by the corresponding weight wt. For an entire generated piece, the weights of each training data fragment could be averaged to obtain the quantitative contribution of that data point (which could then be used, for example, to apportion a revenue share). Our considerations show that the weight distribution is expected to be long-tailed, with a few fragments contributing large weights and many fragments contributing small weights. A suitable cutoff could be used to truncate the number of contributing samples. This process would also show which fragments of the generated piece are truly innovative, since those fragments will not utilize training data fragments with large weights.

[0192] Conclusion

[0193] In this work, we have made the case, with rigorous mathematical proofs as well as in-depth applications in three widely different disciplines (behavioral biology, control engineering, and number theory), that a fundamental rethink of the current toolset of artificial intelligence is possible, where highly parameterized network models are complemented by parameter-free data interpolation. In particular, in some Al applications where an ANN module is currently trained using paired input-output training data, the ANN module could be replaced by a Hilbert kernel. While it is out of the scope of the present work to explore the many ramifications of such an approach, it shows the potential for an entire field of research and applications with a parameter-free interpolation centered approach.

[0194] The potential benefits of replacing hard-to-interpret black-boxes with a theoretically simple and highly transparent approach are manifold. The estimates are straightforward to define, deterministic, and easily reproducible, and do not require any optimization or model selection. This should greatly benefit scientific and biomedical applications by removing the obscuring factors of multi-million parameter black-box networks that investigators have to work hard to design their appropriate architecture and then interpret. It also has ethical-legal utility in permitting proper credit assignment to training data fragments in generative Al, and addressing the concerns of content-creators whose works may be used as training data in generative models.

[0195] In contrast with ANNs, the Hilbert estimator does not require an expensive and lengthy training phase. Nominally it might appear that function evaluation is expensive as it scales linearly with the training set size n. Interestingly, however, current practice in the application of ANNs to very large data sets seems to also follow a linear scaling between the number of model parameters p (and thus the cost of function evaluation) and the training set size n (so called Chinchilla scaling)

[0076] . This is indeed the same scaling as we have in the Hilbert case for evaluation-time compute cost. Notably, in scaling p linearly with / / , a quadratic scaling is induced on the training time compute cost, which may be estimated as p * n assuming that each model data point affects each parameter once during the training phase.

[0196] This enormous cost is simply absent in the Hilbert case - one can start evaluating as soon as the training data set is available and indeed it is trivial to add or subtract data from the training set, in contrast with the ANN case where expensive re-training may need to be carried out. In terms of storage requirements, if the model size scales linearly with the data size, so does the memory requirements, again no different in scaling terms from the Hilbert situation. We summarize this state of affairs in the table below. Note that the training time cost for a large data set is indeed very large - for example if n ~ 1012, then a trillion evaluations would need to occur before the evaluation costs starts exceeding the training costs in the ANN case.

[0197]

[0198] Table 1: Scaling of different items with growing data size.

[0199] So called Chinchilla scaling

[0076] recommends the number of parameters scale with the number of data points with a constant of proportionality ~ 1 / 20. The empirical scaling laws are derived based on the training loss, the scaling of expected risk or test loss is not reported and there is no theoretical basis for a black box network to assume that decrease in training loss implies a decrease in test loss. Hilbert is an interpolation technique and training loss is zero, the expected risk is given by Eq.5. We write O(n) for the evaluation cost and storage as a reminder that using a fixed fraction of the training data does not alter the inverse log scaling of the expected risk.

[0200] The empirically determined scaling laws in current machine learning practice are not based on fundamental theoretical understanding. In these empirical studies, the objective function being optimized is the loss on the training set (ie how well the model fits the data), not the test loss. There is no theoretical reason why training set loss should predict test loss (in fact the double descent phenomena show that in general there is not a simple monotonic relationship between training set loss and test loss). In the Hilbert case the training set loss is zero since training data are interpolated, nevertheless the test loss is proven to go to zero as the training set size increases (the basic result presented in this paper). The striking empirical correspondence between a linear scaling of number of model parameters ("Chinchilla scaling") and the Hilbert case leads us to speculate that the good performance of the large network models studied in

[0076] may indeed be related to data interpolation. The Hilbert framework may indeed provide theoretical background and a comparison point for these and other empirical "scaling laws". This is a promising direction for future research but one well beyond the scope of the current manuscript.

[0201] Note that Hilbert regression is well matched to hardware optimization parallelization as samples may be distributed among threads or cores, and the dot product involved in the distance computation is a vector operation. Further, the sparsification of data by a constant factor, as in the case for "Chinchilla scaling", does not affect the proven convergence rates, and therefore is another way to speed up the evaluation time computation. Such sparsification might be carried out in advance or during evaluation. Given the stark departure from current practice, we fully recognize that many ideas need to be worked out more fully, but there are limitations to what can be done in one paper.

[0202] The success of a purely empirical, data-driven and completely model-free approach in generating predictions with statistically optimal performance raises philosophical questions about scientific understanding. Making good predictions is universally accepted to be a fundamental criterion in judging scientific models, although not the only one. If it is possible to make good predictions simply based on past observations, without models, other criteria such as simplicity and insight, while difficult to quantify, must be brought to the forefront in judging the validity and utility of scientific models of complex phenomena.

[0203] Hilbert transfer function predictor and control

[0204] The prediction process takes as input a series z / t, and generates an output series xt. To

[0205]

[0206] initiate the process, initial conditions (^i ’ -2 >x<i ) and (ui, U2,..,ud-i) have to be specified, following which the values ut are iteratively appended to the input series, and the Tf d

[0207] output at the next time step; Et+i computed using Eq. (10). The initial prediction period (

[0208]

[0209] may be chosen randomly or set to be equal to a previously observed period from the training data.

[0210] During offline prediction, at each time step / , the current input value z / tis appended to the stored input time series -i, -2,.. and the next output ^t+i is predicted using Eq. (10) using a fixed training period of inputs and outputs:

[0211]

[0212] In the online prediction process, each time a predictionX'M is made, the corresponding true value xt+i is appended to the training data set, together with the corresponding control input Ut.

[0213] If acceleration is being predicted using Eq. (10), followed by a double integration to obtain xt+i, the following equations are used to make the prediction (note that finite differences of the training data are used to obtain training values for the predictor variable at):

[0214]

[0215] In FIG. 6a the two-state linear dynamical system used with output Xt and input ut is defined by the equations (in the plots shown, ~ki = ~ki = = 0.1):

[0216] at+\ = klxt + klvt + k3ut, (71)

[0217] vt+1 = vt+ rzt+1, (72)

[0218] xr+l = xt+ vt+1. (73)

[0219] To generate training data, z / twas chosen to be i.i.d with a standard normal distribution. In FIG. 6b the torque-driven, damped pendulum is defined by the discrete-time equations (y = 0.5, dt = 0.2):

[0220] vt+i = vt - (yvt + sin d dt + ut, (74)

[0221] xt+1 = xt + vt+ldt. (75)

[0222] To generate training data, z / twas chosen to be i.i.d with a standard normal distribution. During the control phase, a quadratic cost function in the output C = T

[0223]

[0224] was used.

[0225] For the directed stochastic gradient procedure used to optimize the control cost in the forecast period of a finite time horizon T in MPC, we used the following update procedure

[0070] :

[0226]

[0227] Here Ut(f) = (ub ub+i,.. u't+T-i) denotes a trial control vector of length, which is iteratively updated for a number of Monte Carlo iterations i = 1.. M (we chose M= 100). C(X(U), U) is the ( / -dependent cost function, which is computed by first predicting the output vectorl't+ 1 1C), x+T((■'), either using the Hilbert procedure, or the true system model as a baseline. Note, that in this case the offline Hilbert procedure has to be used, since these are Monte Carlo iterations and the true value of the output is not known.

[0228] For the first iteration, for the first time step, U1 is initialized randomly. For subsequent time steps, The first T~1 components of the vector C7i( / +1) are initialized using last T~ 1 components of Uu(f). The stochastic steps dUi are chosen to be random directions on the T-dimensional unit sphere. The gain g was chosen to be 10 and the step size e = 0.3 in the control applications shown in FIG. 7.

[0229] After M stochastic directional gradient steps, and following the MPC procedure, we take only the first time step Ut from the vector U (f) and apply this as the control input. We then observe the real system output xt+i, and append both / / < and xt+i to the training data before iterating the MPC loop.

[0230] Extended Data Figures

[0231] FIG. 10 - Hilbert classification (N=1000): P(1|x)~N([0,2],1) P(-1,x)=N([0,0],1: Classification using the Hilbert kernel: A simple example is shown, with two classes of points drawn from a mixture of 2D unit Normal distributions, with mean separated by 2. The points are shown in green and red colors (1000 points of each class). The red vertical line is the Bayes classification boundary. The yellow and blue colored regions are the Hilbert-predicted classification regions for the green and red points. The islands of blue in yellow (and vice versa) are due to the interpolative nature of the classifier, and correspond to the phenomenon of adversarial examples which are guaranteed for interpolating classifiers on noisy data.

[0232] FIG. 11: Behavior of 2 Hilbert fish without the tank wall. This figure is the analog of FIG. 3 in the main text (also for a memory M = 2), but in the case where the presence of the tank wall is not enforced in the Hilbert model. The different panels show the 9 observables used to characterize the individual (a-c) and collective (d-f) behavior, and the time correlations in the system (g-i): a, PDF of the speed, V b, PDF of the distance to the wall, rw; c, PDF of the heading angle relative to the normal to the wall, 0W; d, PDF of the distance between the pair of individuals, d e, PDF of the group polarization,p= cos(A<^ / 2)(where A<) is the relative heading angle; f, PDF of the viewing angle at which an individual perceives the other individual, / . See FIG. 3a and b in the main text for a visual representation of the main variables, g, Mean squared displacement, Cx(f), and its asymptotic limit, Cx(co) = 2(r2) (dotted lines); h, Velocity autocorrelation, C( / ); i, Polarization autocorrelation, Cp(f). The black PDFs correspond to experiments, while the red PDFs correspond to the predictions of the Hilbert generative model. The plots are on the same scale as in FIG. 3 in the main text, except for rw, for which the horizontal axis has been extended to negative values of rwcorresponding to instances where an individual is observed outside the limits of the experimental circular tank. Yet, the Hilbert fish spend 87% of the time strictly within the tank limits, and when they wander outside the tank, their average excursion distance from the wall is only 1.3cm. These excursions are responsible for the upward and rightward shift of the peak of Cx(f) and for the larger asymptotic limit, Cx(co) = 2{r2) ~ 980cm2(compared to Gc(oo) ~ 900cm2for fish or for the Hilbert model implementing the rejection procedure enforcing the presence of the tank wall).

[0233] FIG. 12: Entropy time series for 2 Hilbert fish. We plot a 2-minute time series of the entropy for 2 Hilbert fish, and for a memory M = 2. The entropy S can be interpreted as log2 N, where N is the effective number of real fish configurations used to predict the acceleration of the Hilbert fish. The time series exhibits short periods where N ~ 1 (5 ~ 0), when the Hilbert scheme has essentially selected a unique real fish configuration (“copying”). This short time series also presents three short periods when N > 64 (5 > 6). The PDF of the entropy computed over much longer time is shown in FIG. 13.

[0234] FIG. 13: Entropy distribution for 2 Hilbert fish. We plot the PDF of the entropy for 2 Hilbert fish (for a memory M = 2) resulting from an effective simulation time of 3 hours. The PDF of the entropy is reasonably well fitted by the normalized functional form

[0235]

[0236] although the fit does not capture some

[0237] outliers at S > 10. The fitted cut-off entropy scale, Sc~ 3.61, corresponds to Nc= 2Sc~ 12.2 real fish configurations contributing to the acceleration prediction, while the mean entropy (5) ~ 1.37 corresponds to 2<log2 N> = 2<s)~ 2.6 configurations. The mean number of configurations used for a prediction is (N) = (2s), and is dominated by outliers. If this average is restrained to instance where S < 10, one finds (N)s<io~ 5.2 (our fit p(S) would predict (N)s<io ~ 4.7), whereas the average including all data is (N) ~ 35. Also note the small peak in the PDF near 5= 1, corresponding to N = 2 relevant configurations contributing almost equally to the Hilbert prediction. Yet, during the simulation, entropies as high as S ~ 15 were recorded, corresponding to N ~ 32768 fish configurations effectively considered by the Hilbert kernel. Compared to ANN methods, the Hilbert interpolation scheme is hence able to adapt the effective number of used data for the prediction to the properties of the input vector. See also FIG. 12 for a short time series of the entropy.

[0238] FIG. 14: Autoregressive generative modeling of time series data: three examples are shown of signals generated by Eq. (9) with simple training data consisting of N= 10000 samples of a single training signal. The three columns respectively show results corresponding to a training signal generated by white Gaussian noise, a sinusoid, and iterates of a logistic regression equation xn+1= Axn(1-xn) for A = 3.9. A lag window size of T= 10 is used (see the next figure for a lag window size of T= 40, and the signal generation is initiated by random initial conditions consisting of T samples of a standard normal distributed variable. The second row shows the generated signal. The third row shows the entropy of the generative weights as a function of generation time, and the fourth row shows the position of the maximum weight in the training signal, also as a function of generation time. Note the “copying” behavior, where the generated signal starts some fragment of the training signal after an initial transient. During the “copying” phase, the weight entropy falls to zero, and the index of the maximum weight increments linearly with time.

[0239] FIG. 15: Autoregressive generative modeling of time series data: the same signal examples as in the previous extended data figure (FIG. 14) are shown, this time with a lag window T= 40. In this case, periodic behavior is observed after an initial transient. The periods themselves contain short episodes of “copying” where the entropy falls to zero, and also show short episodes of close to fixed-point behavior of the dynamics, where the generated signal has an almost constant value. The extent of the “copying”, periodic or constant behaviors depends on the initial conditions as well as on d.

[0240] FIG. 16: Impact of an initial coordinate transformation x — <b(x). This example shows two Hilbert kernel fits, with the black curve corresponding to the original kernel, and the blue curve corresponding to the generalized kernel corresponding to replacing x — <b(x) in the weights. The samples Xi (z = 1..100) are chosen so that the transformed coordinates E>(x / ) have a uniform distribution. The red line corresponds toy = x and uncorrelated Gaussian noise with o = 0.1 is added to produce the noisy samples. The two regression functions both interpolate, but show slight differences, especially in the data-sparse region, with the transformed weights (that produce uniform sampling of x) being a bit closer to the noise-free function. As proven in the paper, both estimates are statistically consistent and have the same large-sample asymptotics in the leading order, but the sub-leading order behavior will generally depend on in conjunction with the other details of the problem. References

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[0318] MISCELLANEOUS

[0319] Various embodiments of the invention may be implemented at least in part in any conventional computer programming language. For example, some embodiments may be implemented in a procedural programming language (e.g., “C”), or in an object-oriented programming language (e.g., “C++”). Other embodiments of the invention may be implemented as a pre-configured, stand-alone hardware element and / or as preprogrammed hardware elements (e.g., application specific integrated circuits, FPGAs, and digital signal processors), or other related components.

[0320] In alternative embodiments, the disclosed apparatus and methods (e.g., as in any flow charts or logic flows described above) may be implemented as a computer program product for use with a computer system. Such implementation may include a series of computer instructions fixed on a tangible, non-transitory medium, such as a computer readable medium (e.g., a diskette, CD-ROM, ROM, or fixed disk). The series of computer instructions can embody all or part of the functionality previously described herein with respect to the system.

[0321] Those skilled in the art should appreciate that such computer instructions can be written in a number of programming languages for use with many computer architectures or operating systems. Furthermore, such instructions may be stored in any memory device, such as a tangible, non-transitory semiconductor, magnetic, optical or other memory device, and may be transmitted using any communications technology, such as optical, infrared, RF / microwave, or other transmission technologies over any appropriate medium, e.g., wired (e.g., wire, coaxial cable, fiber optic cable, etc.) or wireless (e.g., through air or space).

[0322] Among other ways, such a computer program product may be distributed as a removable medium with accompanying printed or electronic documentation (e.g., shrink wrapped software), preloaded with a computer system (e.g., on system ROM or fixed disk), or distributed from a server or electronic bulletin board over the network (e.g., the Internet or World Wide Web). In fact, some embodiments may be implemented in a software-as-a-service model (“SAAS”) or cloud computing model. Of course, some embodiments of the invention may be implemented as a combination of both software (e.g., a computer program product) and hardware. Still other embodiments of the invention are implemented as entirely hardware, or entirely software.

[0323] Computer program logic implementing all or part of the functionality previously described herein may be executed at different times on a single processor (e.g., concurrently) or may be executed at the same or different times on multiple processors and may run under a single operating system process / thread or under different operating system processes / threads. Thus, the term “computer process” refers generally to the execution of a set of computer program instructions regardless of whether different computer processes are executed on the same or different processors and regardless of whether different computer processes run under the same operating system process / thread or different operating system processes / threads. Software systems may be implemented using various architectures such as a monolithic architecture or a microservices architecture.

[0324] It should be noted that terms such as “computer” or “processor” may be used herein to describe devices or systems that may be used in certain embodiments of the present invention and should not be construed to limit the present invention to any particular device or system type unless the context otherwise requires. Such devices or systems typically include one or more network interfaces for communicating over a communication network and at least one processor (e.g., a microprocessor with memory and other peripherals and / or applicationspecific hardware) configured accordingly to perform device or system functions.

[0325] Communication networks generally may include public and / or private networks; may include local-area, wide-area, metropolitan-area, storage, and / or other types of networks; and may employ communication technologies including, but in no way limited to, analog technologies, digital technologies, optical technologies, wireless technologies (e.g., Bluetooth, WiFi, cellular, etc.), networking technologies, and internetworking technologies.

[0326] It should also be noted that devices and systems may use communication protocols and messages (e.g., messages created, transmitted, received, stored, and / or processed by the device or system), and such messages may be conveyed by a communication network or medium. Unless the context otherwise requires, the present invention should not be construed as being limited to any particular communication message type, communication message format, or communication protocol. Thus, a communication message generally may include, without limitation, a frame, packet, datagram, user datagram, cell, or other type of communication message. Unless the context requires otherwise, references to specific communication protocols are exemplary, and it should be understood that alternative embodiments may, as appropriate, employ variations of such communication protocols (e.g., modifications or extensions of the protocol that may be made from time-to-time) or other protocols either known or developed in the future.

[0327] It should also be noted that logic flows may be described herein to demonstrate various aspects of the invention, and should not be construed to limit the present invention to any particular logic flow or logic implementation. The described logic may be partitioned into different logic blocks (e.g., programs, modules, functions, or subroutines) without changing the overall results or otherwise departing from the true scope of the invention. Often times, logic elements may be added, modified, omitted, performed in a different order, or implemented using different logic constructs (e.g., logic gates, looping primitives, conditional logic, and other logic constructs) without changing the overall results or otherwise departing from the true scope of the invention.

[0328] The present invention may be embodied in many different forms, including, but in no way limited to, computer program logic for use with a processor (e.g., a microprocessor, microcontroller, digital signal processor, or general purpose computer), programmable logic for use with a programmable logic device (e.g., a Field Programmable Gate Array (FPGA) or other PLD), discrete components, integrated circuitry (e.g., an Application Specific Integrated Circuit (ASIC)), or any other means including any combination thereof. Computer program logic implementing some or all of the described functionality is typically implemented as a set of computer program instructions that is converted into a computer executable form, stored as such in a computer readable medium, and executed by one or more processors optionally under the control of an operating system. Hardware-based logic implementing some or all of the described functionality may be implemented using one or more appropriately configured FPGAs or other programmable logic devices.

[0329] Computer program logic implementing all or part of the functionality previously described herein may be embodied in various forms, including, but in no way limited to, a source code form, a computer executable form, and various intermediate forms (e.g., forms generated by an assembler, compiler, linker, or locator). Source code may include a series of computer program instructions implemented in any of various programming languages (e.g., an object code, an assembly language, or a high-level language such as Fortran, C, C++, JAVA, Python, or HTML) for use with various operating systems or operating environments. The source code may define and use various data structures and communication messages. The source code may be in a computer executable form (e.g., via an interpreter), or the source code may be converted (e.g., via a translator, assembler, or compiler) into a computer executable form.

[0330] Computer program logic implementing all or part of the functionality previously described herein may be executed at different times on a single processor (e.g., concurrently) or may be executed at the same or different times on multiple processors and may run under a single operating system process / thread or under different operating system processes / threads. Thus, the term “computer process” refers generally to the execution of a set of computer program instructions regardless of whether different computer processes are executed on the same or different processors and regardless of whether different computer processes run under the same operating system process / thread or different operating system processes / threads.

[0331] The computer program may be fixed in any form (e.g., source code form, computer executable form, or an intermediate form) either permanently or transitorily in a tangible storage medium, such as a semiconductor memory device (e.g., a RAM, ROM, PROM, EEPROM, or Flash-Programmable RAM), a magnetic memory device (e.g., a diskette or fixed disk), an optical memory device (e.g., a CD-ROM), a PC card (e.g., PCMCIA card), or other memory device. The computer program may be fixed in any form in a signal that is transmittable to a computer using any of various communication technologies, including, but in no way limited to, analog technologies, digital technologies, optical technologies, wireless technologies (e.g., Bluetooth), networking technologies, and internetworking technologies. The computer program may be distributed in any form as a removable storage medium with accompanying printed or electronic documentation (e.g., shrink wrapped software), preloaded with a computer system (e.g., on system ROM or fixed disk), or distributed from a server or electronic bulletin board over the communication system (e.g., the Internet or World Wide Web).

[0332] Hardware logic (including programmable logic for use with a programmable logic device) implementing all or part of the functionality previously described herein may be designed using traditional manual methods, or may be designed, captured, simulated, or documented electronically using various tools, such as Computer Aided Design (CAD), a hardware description language (e.g., VHDL or AHDL), or a PLD programming language (e.g., PALASM, ABEL, or CUPL).

[0333] Programmable logic may be fixed either permanently or transitorily in a tangible storage medium, such as a semiconductor memory device (e.g., a RAM, ROM, PROM, EEPROM, or Flash-Programmable RAM), a magnetic memory device (e.g., a diskette or fixed disk), an optical memory device (e.g., a CD-ROM), or other memory device. The programmable logic may be fixed in a signal that is transmittable to a computer using any of various communication technologies, including, but in no way limited to, analog technologies, digital technologies, optical technologies, wireless technologies (e.g., Bluetooth), networking technologies, and internetworking technologies. The programmable logic may be distributed as a removable storage medium with accompanying printed or electronic documentation (e.g., shrink wrapped software), preloaded with a computer system (e.g., on system ROM or fixed disk), or distributed from a server or electronic bulletin board over the communication system (e.g., the Internet or World Wide Web). Of course, some embodiments of the invention may be implemented as a combination of both software (e.g., a computer program product) and hardware. Still other embodiments of the invention are implemented as entirely hardware, or entirely software.

[0334] While the invention has been particularly shown and described with reference to specific embodiments, it will be understood by persons of ordinary skill in the art that various changes in form and detail may be made without departing from the spirit and scope of the invention as defined by the appended clauses. While some of these embodiments have been described in the claims by process steps, an apparatus comprising a computer capable of executing the process steps is also included in the present invention. Likewise, a computer program product comprising a tangible, non-transitory computer readable medium having embodied therein computer executable instructions for executing the process steps is included in the present invention. Data signals embodying computer program instructions and / or messages received or transmitted over a communication system are also included in the present invention. Unless the context requires otherwise, the various functions and features described herein can be used in combination even if disclosed or claimed individually. Thus, for example, it is contemplated that dependent claims included below could be rewritten into multiple dependent form to depend from the base claim and an intervening claim(s).

[0335] Importantly, it should be noted that embodiments of the present invention may employ conventional components such as conventional computers (e.g., off-the-shelf PCs, mainframes, microprocessors), conventional programmable logic devices (e.g., off-the shelf FPGAs or PLDs), or conventional hardware components (e.g., off-the-shelf ASICs or discrete hardware components) which, when programmed or configured to perform the non-conventional methods described herein, produce non-conventional devices or systems. Thus, there is nothing conventional about the inventions described herein because even when embodiments are implemented using conventional components, the resulting devices and systems are necessarily non-conventional because, absent special programming or configuration, the conventional components do not inherently perform the described non-conventional functions.

[0336] The activities described and claimed herein provide technological solutions to problems that arise squarely in the realm of technology. These solutions as a whole are not well-understood, routine, or conventional and in any case provide practical applications that transform and improve computers and computer routing systems.

[0337] While various inventive embodiments have been described and illustrated herein, those of ordinary skill in the art will readily envision a variety of other means and / or structures for performing the function and / or obtaining the results and / or one or more of the advantages described herein, and each of such variations and / or modifications is deemed to be within the scope of the inventive embodiments described herein. More generally, those skilled in the art will readily appreciate that all parameters, dimensions, materials, and configurations described herein are meant to be exemplary and that the actual parameters, dimensions, materials, and / or configurations will depend upon the specific application or applications for which the inventive teachings is / are used. Those skilled in the art will recognize, or be able to ascertain using no more than routine experimentation, many equivalents to the specific inventive embodiments described herein. It is, therefore, to be understood that the foregoing embodiments are presented by way of example only and that, within the scope of the appended claims and equivalents thereto, inventive embodiments may be practiced otherwise than as specifically described and claimed. Inventive embodiments of the present disclosure are directed to each individual feature, system, article, material, kit, and / or method described herein. In addition, any combination of two or more such features, systems, articles, materials, kits, and / or methods, if such features, systems, articles, materials, kits, and / or methods are not mutually inconsistent, is included within the inventive scope of the present disclosure.

[0338] Various inventive concepts may be embodied as one or more methods, of which examples have been provided. The acts performed as part of the method may be ordered in any suitable way. Accordingly, embodiments may be constructed in which acts are performed in an order different than illustrated, which may include performing some acts simultaneously, even though shown as sequential acts in illustrative embodiments. All definitions, as defined and used herein, should be understood to control over dictionary definitions, definitions in documents incorporated by reference, and / or ordinary meanings of the defined terms.

[0339] The indefinite articles “a” and “an,” as used herein in the specification and in the claims, unless clearly indicated to the contrary, should be understood to mean “at least one.” The phrase “and / or,” as used herein in the specification and in the claims, should be understood to mean “either or both” of the elements so conjoined, i.e., elements that are conjunctively present in some cases and disjunctively present in other cases. Multiple elements listed with “and / or” should be construed in the same fashion, i.e., “one or more” of the elements so conjoined. Other elements may optionally be present other than the elements specifically identified by the “and / or” clause, whether related or unrelated to those elements specifically identified. Thus, as a non-limiting example, a reference to “A and / or B”, when used in conjunction with open-ended language such as “comprising” can refer, in one embodiment, to A only (optionally including elements other than B); in another embodiment, to B only (optionally including elements other than A); in yet another embodiment, to both A and B (optionally including other elements); etc.

[0340] As used herein in the specification and in the claims, “or” should be understood to have the same meaning as “and / or” as defined above. For example, when separating items in a list, “or” or “and / or” shall be interpreted as being inclusive, i.e., the inclusion of at least one, but also including more than one, of a number or list of elements, and, optionally, additional unlisted items. Only terms clearly indicated to the contrary, such as “only one of’ or “exactly one of,” or, when used in the claims, “consisting of,” will refer to the inclusion of exactly one element of a number or list of elements. In general, the term “or” as used herein shall only be interpreted as indicating exclusive alternatives (i.e., “one or the other but not both”) when preceded by terms of exclusivity, such as “either,” “one of,” “only one of,” or “exactly one of.” “Consisting essentially of,” when used in the claims, shall have its ordinary meaning as used in the field of patent law.

[0341] As used herein in the specification and in the claims, the phrase “at least one,” in reference to a list of one or more elements, should be understood to mean at least one element selected from any one or more of the elements in the list of elements, but not necessarily including at least one of each and every element specifically listed within the list of elements and not excluding any combinations of elements in the list of elements. This definition also allows that elements may optionally be present other than the elements specifically identified within the list of elements to which the phrase “at least one” refers, whether related or unrelated to those elements specifically identified. Thus, as a non-limiting example, “at least one of A and B” (or, equivalently, “at least one of A or B,” or, equivalently “at least one of A and / or B”) can refer, in one embodiment, to at least one, optionally including more than one, A, with no B present (and optionally including elements other than B); in another embodiment, to at least one, optionally including more than one, B, with no A present (and optionally including elements other than A); in yet another embodiment, to at least one, optionally including more than one, A, and at least one, optionally including more than one, B (and optionally including other elements); etc.

[0342] As used herein in the specification and in the claims, all transitional phrases such as “comprising,” “including,” “carrying,” “having,” “containing,” “involving,” “holding,” “composed of,” and the like are to be understood to be open-ended, i.e., to mean including but not limited to. Only the transitional phrases “consisting of’ and “consisting essentially of’ shall be closed or semi-closed transitional phrases, respectively, as set forth in the United States Patent Office Manual of Patent Examining Procedures, Section 2111.03.

[0343] It should be noted that connecting lines with or without arrows may be used in drawings to represent communication, transfer, or other activity involving two or more entities. Connecting lines with double-ended arrows generally indicate that activity may occur in both directions (e.g., a command / request in one direction with a corresponding reply back in the other direction, or peer-to-peer communications initiated by either entity), although in some situations, activity may not necessarily occur in both directions.

[0344] Connecting lines with single-ended arrows generally indicate activity exclusively or predominantly in the direction of the arrow, although it should be noted that, in certain situations, such directional activity may involve activities in the opposite direction or in both directions (e.g., a message from a sender to a receiver and an acknowledgment back from the receiver to the sender, or establishment of a connection prior to a transfer and termination of the connection following the transfer). Thus, the type of arrow used in a particular drawing to represent a particular activity is exemplary and should not be seen as limiting. Connecting lines with no arrows can indicate activity in one, the other, or both directions as the context suggests. Dashed connecting lines may be used to represent optional or ancillary activities as the context suggests.

[0345] Various embodiments of the present invention may be characterized by the potential claims listed in the paragraphs following this paragraph (and before the actual claims provided at the end of the application). These potential claims form a part of the written description of the application. Accordingly, subject matter of the following potential claims may be presented as actual claims in later proceedings involving this application or any application claiming priority based on this application. Inclusion of such potential claims should not be construed to mean that the actual claims do not cover the subject matter of the potential claims. Thus, a decision to not present these potential claims in later proceedings should not be construed as a donation of the subject matter to the public. Nor are these potential claims intended to limit various pursued claims.

[0346] Without limitation, potential subject matter that may be claimed (prefaced with the letter “P” so as to avoid confusion with the actual claims presented below) includes:

[0347] Pl. A method of credit assignment in Al where the relative contribution of individual training data elements to the output of the system is determined by a non-negative weight, where the weights sum to one, and where the weights are only dependent on the training data and not dependent on trainable or tunable model parameters.

[0348] P2. A business method in which a creator or other data source (writers, musicians, artists or other sources) who contributes to training data sets for an Al system, can be compensated proportionally to the contribution of the training data element contributed by that creator or other data source in a transparent manner not subjected to arbitrary black-box parameters, optionally wherein weights such as the weights in claim 1 are used to determine and distribute the relative compensation among the pool of creators.

[0349] P3. A generative model for language, music, art, or other artifacts generated from a training data corpus, where the generative process is transparent and parameter free, and the relative contribution of the input training data to the generated output can be determined in a transparent manner in terms of non-zero weights that sum to one.

[0350] P4. An example of P3 above is a generalized Hilbert-kemel based autoregressive model of time series data, that can be used to generate language or music output, or a generalized Hilbert-kemel based diffusion or flow model, where the diffusion or flow process is governed by parameter-free weights such as the Hilbert weight.

[0351] P5. A search engine or a language model with data provenance, where the precise contribution of individual elements being searched over, to the output of the search engine or the language model, can be determined transparently in terms of data-dependent non-negative credit-assignment weights that are not determined by tuning model parameters of a complex network model, optionally wherein the weights range between zero and one and sum to one. P6. A parameter-free method for making predictions for the financial markets based on the Hilbert generative modeling approach.

[0352] P7. A transparent method for determining information provenance, in a legal application where a summary or other output is created based on a corpus of training data such as legal documents from the past.

[0353] P8. A method for a-posteriori determination of credit assignment for output of a network-based generative model, such as a large language model or a diffusion model for image, music or video generation, based on applying the Hilbert weighting scheme to the output of such black box models.

[0354] P9. A method for data compression in which past data points in a series are used to predict the next data point, using the same parameter free predictive approach as in the above claims, and in which only the difference between the actual value and the predicted value is stored (the prediction error). If there is predictability in the process, then as past data points accumulate, the prediction error becomes smaller and smaller, and can therefore be stored more efficiently, leading to data compression. During the decompression or decoding phase, the prediction error series can be cumulatively added in order to recover the original series. The data being compressed could include audio or video data. By way of background, data compression is a universally important task, since the storage and transmission of large volumes of data can be cost prohibitive and resource consuming. Data compression includes two steps, a coder that takes as an input the original data set and produces an output, compressed data set, and a decoder that takes the compressed data set and reproduces the original data set (in case of lossless compression) or a close approximation (in the case of lossy compression) of the original data set. Many data compression methods exist, for example the Lempel-Ziv algorithm. Many of those compression methods utilize parameters in the process of data compression, including new methods based on ANNs. We propose a parameter-free data compression scheme, in which the coding step involves predicting a signal or series from its past values utilizing the Hilbert approach as above, computing the residuals, and storing only the residuals in the compressed format. If there is predictability to the series, which is almost always the case since there will be some degree of structure to the data, the residuals will be smaller than the original data series, and hence the data will be compressed.

[0355] PIO. A method for feedback control of systems, in which predictions are derived from the generative modeling approach above. Note that in conventional feedback control systems, the predictions are derived from a parametric system model which is learned from data. The purely data derived prediction scheme would replace the parametric system model used for feedback control. By way of background, feedback control of systems is central to modern technology, from the household thermostat controlling heating and cooling to industrial automation to robotics. In feedback control, data derived from a set of sensors (such as positions or angles in robotics) is passed through a controller or control algorithm, to generate actuator signals (such as motor drive commands), with the objective being to keep certain outputs, such as the position or behavior of the sensed object, close to some desired position or behavior. Central to the feedback control approach is the control algorithm, which converts the sensor signals to actuator signals. This control algorithm often utilizes parametric models of the system under consideration to predict the future behavior of the system based on the current sensor values and the actuator signals that the control algorithm is generating. For example, currently, ANNs are often being considered to create such parametric models. We propose to replace one or more models utilized in such a control system by the Hilbert prediction scheme described above. Examples of such applications have been provided in the attached papers, e.g., for stabilizing an inverted pendulum or a chaotic map around an unstable fixed point. The benefit of this approach is that the system can adapt on the fly, without having to choose and train a parametric model.

[0356] Pl 1. We intend to claim broad applications to artificial intelligence or machine learning systems, where a core component of the system includes the learning of a function from data. Instead of using artificial neural networks or other parametric models for learning this core function, we will utilize the predictive framework described in the above claims. By way of background, modern Al or ML systems usually contain one or more core components, which are input-output functions, where the function converts certain input data to certain output data. In training or optimizing these systems, training data sets are obtained containing pairs of input and output data values. Often, parametric models are utilized to implement such functions, with these parametric models being ANNs in many modern applications. The parameters of these models are then chosen based on an optimization algorithm (“learning” or “training”), based on the training data set. Our approach is fundamentally different in that it does not have parameters and does not need a parameter optimization step once the data representation has been chosen. The training set is directly utilized in the Hilbert kernel approach we propose to evaluate the function outputs. Although the above discussion discloses various exemplary embodiments of the invention, it should be apparent that those skilled in the art can make various modifications that will achieve some of the advantages of the invention without departing from the true scope of the invention. Any references to the “invention” are intended to refer to exemplary embodiments of the invention and should not be construed to refer to all embodiments of the invention unless the context otherwise requires. The described embodiments are to be considered in all respects only as illustrative and not restrictive.

Claims

What is claimed is:

1. A computer-implemented method for assigning credit to artificial intelligence training items, the method comprising:receiving a list of a plurality of training items and a new item of a same type as the plurality of training items;associating a numerical vector of a pre-determined dimension to each training item as well as to the new item, wherein the pre-determined dimension is the same for each training item and the new item;computing, for each training data item, a non-negative weight quantifying resemblance of the new item to the training data item, where such weights are specified by a predetermined function of training data vectors and the new vector, and which does not involve any optimization or tunable parameters at any time during the credit assignment process; andassigning credit to at least one of the training items based on the weights.

2. The method of claim 1, wherein credit is assigned to the training items proportionally to the weights.

3. The method of claim 1, wherein the weight associated with a given training data item is proportional to an inverse power of a distance between the new vector and the training data vector.

4. The method of claim 3, wherein the power is given by the dimension of the vectors.

5. The method of claim 1, wherein the training items are fragments of information of a given type, and wherein the new item is a fragment of the given type, and the weights are used to determine the relative contribution of the training items to the new item.

6. The method of claim 5, wherein the given type is one of audio (e.g., music), image, video, text (e.g., word tokens), numerical, financial, or medical information.

7. The method of claim 5, wherein the weights are used to determine copyright infringement in materials generated by humans or by Al applications.

8. The method of claim 5, wherein the weights are used to determine plagiarism in materials generated by humans or by Al applications.

9. The method of claim 5, wherein the plurality of training items are items among which a search query is performed, the query input constitutes the new item, and the weights for each input item are used to assign scores to the input items returned by the search.

10. The method of claim 1, wherein:the training items include training input and output pairs, for which, along with the new input, an output prediction is desired; andassigning credit to at least one of the training items based on the weights comprises computing the desired prediction by combining the training outputs and their associated weights.

11. The method of claim 10, wherein the prediction is statistically consistent.

12. The method of claim 10, wherein the weight associated with a training data input is proportional to an inverse power of the distance between the new input vector and a training input vector.

13. The method of claim 12, wherein the power is given by the dimension of the input vectors.

14. The method of claim 10, wherein each item of the training input data is a fixed length segment of a training time series, the corresponding training output is the next term in the training time series, the new input vector is a segment of the same type of time series for which a prediction of next term is desired.

15. The method of claim 14, wherein the time series consists of a sequence of audio samples, or a sequence of sub-images of an image, or a sequence of video frames, or asequence of text (e.g., word tokens) suitable for language modeling, or a sequence of financial information, or a sequence of medical information.

16. The method of claim 1, wherein the credit includes compensation for a contributor of a training item.

17. The method of claim 16, wherein assigning credit to at least one of the training items based on the weights comprises assigning compensation to at least one of the training items based on the weights.

18. The method of claim 17, wherein compensation is assigned proportionally to the weights.

19. The method of claim 17, wherein the weight associated with a training data item is proportional to an inverse power of the distance between the new vector and a training data vector.

20. The method of claim 19, wherein the power is given by the dimension of the vectors.

21. The method of claim 17, wherein the training items are fragments of information of a given type, and wherein the new item is a fragment of the given type.

22. The method of claim 21, wherein the given type is one of audio, video, or text.

23. The method of claim 17, wherein a total compensation associated with an individual training item, is determined for a collection of new items, by separately computing the weights corresponding to each of the new items and the individual training item, the weights being normalized to unity for each new item, then adding those weights.

24. A system for assigning credit to artificial intelligence training items, the system comprising:at least one processor and at least one tangible, non-transitory computer readable medium having embodied therein computer program instructions which, when executed using the at least one processor, perform computer processes comprising:receiving a list of a plurality of training items and a new item of a same type as the plurality of training items;associating a numerical vector of a pre-determined dimension to each training item as well as to the new item, wherein the pre-determined dimension is the same for each training item and the new item;computing, for each training data item, a non-negative weight quantifying resemblance of the new item to the training data item, where such weights are specified by a predetermined function of training data vectors and the new vector, and which does not involve any optimization or tunable parameters at any time during the credit assignment process; andassigning credit to at least one of the training items based on the weights.

25. The system of claim 24, wherein credit is assigned to the training items proportionally to the weights.

26. The system of claim 24, wherein the weight associated with a given training data item is proportional to an inverse power of a distance between the new vector and the training data vector.

27. The system of claim 26, wherein the power is given by the dimension of the vectors.

28. The system of claim 24, wherein the training items are fragments of information of a given type, and wherein the new item is a fragment of the given type, and the weights are used to determine the relative contribution of the training items to the new item.

29. The system of claim 28, wherein the given type is one of audio (e.g., music), image, video, text (e.g., word tokens), numerical, financial, or medical information.

30. The system of claim 28, wherein the weights are used to determine copyright infringement in materials generated by humans or by Al applications.

31. The system of claim 28, wherein the weights are used to determine plagiarism in materials generated by humans or by Al applications.

32. The system of claim 28, wherein the plurality of training items are items among which a search query is performed, the query input constitutes the new item, and the weights for each input item are used to assign scores to the input items returned by the search.

33. The system of claim 24, wherein:the training items include training input and output pairs, for which, along with the new input, an output prediction is desired; andassigning credit to at least one of the training items based on the weights comprises computing the desired prediction by combining the training outputs and their associated weights.

34. The system of claim 33, wherein the prediction is statistically consistent.

35. The system of claim 33, wherein the weight associated with a training data input is proportional to an inverse power of the distance between the new input vector and a training input vector.

36. The system of claim 35, wherein the power is given by the dimension of the input vectors.

37. The system of claim 33, wherein each item of the training input data is a fixed length segment of a training time series, the corresponding training output is the next term in the training time series, the new input vector is a segment of the same type of time series for which a prediction of next term is desired.

38. The system of claim 37, wherein the time series consists of a sequence of audio samples, or a sequence of video frames, or a sequence of word tokens suitable for language modeling, or a sequence of financial information.

39. The system of claim 24, wherein the credit includes compensation for a contributor of a training item.

40. The system of claim 39, wherein assigning credit to at least one of the training items based on the weights comprises assigning compensation to at least one of the training items based on the weights.

41. The system of claim 40, wherein compensation is assigned proportionally to the weights.

42. The system of claim 40, wherein the weight associated with a training data item is proportional to an inverse power of the distance between the new vector and a training data vector.

43. The system of claim 42, wherein the power is given by the dimension of the vectors.

44. The system of claim 40, wherein the training items are fragments of information of a given type, and wherein the new item is a fragment of the given type.

45. The system of claim 44, wherein the given type is one of audio, video, or text.

46. The system of claim 40, wherein a total compensation associated with an individual training item, is determined for a collection of new items, by separately computing the weights corresponding to each of the new items and the individual training item, the weights being normalized to unity for each new item, then adding those weights.

47. A computer program product comprising at least one tangible, non-transitory computer readable medium having embodied therein computer program instructions for assigning credit to artificial intelligence training items which, when executed using at least one computer processor, perform computer processes comprising:receiving a list of a plurality of training items and a new item of a same type as the plurality of training items;associating a numerical vector of a pre-determined dimension to each training item as well as to the new item, wherein the pre-determined dimension is the same for each training item and the new item;computing, for each training data item, a non-negative weight quantifying resemblance of the new item to the training data item, where such weights are specified by a predetermined function of training data vectors and the new vector, and which does not involve any optimization or tunable parameters at any time during the credit assignment process; andassigning credit to at least one of the training items based on the weights.

48. A computer implemented method for generating predictions, comprising:receiving a list of training input and output pairs, and a new input, for which an output prediction is desired;associating numerical vectors of the same pre-determined dimension to each training input item as well as the new input item;computing a non-negative weight for each training input quantifying its resemblance to the new input, where such weights are specified by a predetermined function of training data vectors and the new vector, and which does not involve any optimization or tunable parameters; andcomputing the desired prediction by combining the training outputs and their associated weights.

49. The method of claim 48, wherein the prediction is statistically consistent.

50. The method of claim 48, wherein the weight associated with a training data input is proportional to an inverse power of the distance between the new input vector and a training input vector.

51. The method of claim 48, wherein each item of the training input data is a fixed length segment of a training time series, the corresponding training output is the next term in the training time series, the new input vector is a segment of the same type of time series for which a prediction of next term is desired.

52. The method of claim 51, wherein the time series consists of a sequence of audio samples, or a sequence of video frames, or a sequence of word tokens suitable for language modeling, or a sequence of financial information.

53. The method of claim 50, wherein the power is given by the dimension of the input vectors.

54. A computer-implemented method for compensating contributors to training data for generative Al, comprising:receiving a list of training items and a new item of the same type;associating a numerical vector of the same pre-determined dimension to each training item as well as the new item;computing non-negative weights quantifying the resemblance of the new item to each training data item, where such weights are specified by a predetermined function of the training data vectors and the new vector, and which does not involve any optimization or tunable parameters at any time during the process; andassigning compensation proportionally to these weights.

55. The method of claim 54, wherein the weight associated with a training data item is proportional to an inverse power of the distance between the new vector and a training data vector.

56. The method of claim 55, wherein the power is given by the dimension of the vectors.

57. The method of claim 54, wherein the training items are fragments of music audio, or video, or text, and the new item is a fragment of the same type.

58. The method of claim 54, wherein a total compensation associated with an individual training item, is determined for a collection of new items, by separately computing the weights corresponding to each of the new items and the individual training item, the weights being normalized to unity for each new item, then adding those weights.

59. A computer implemented method for sequence generation with provenance, comprising:receiving a set of training input sequences;creating a set of training input-output pairs, where the training inputs are fragments of the input sequence of fixed length T, and the training output for each such fragment is the sequence element immediately following the said fragment;associating numerical vectors of the same pre-determined dimension to each training input item;initiating the sequence generation process with a starting fragment of the same length; computing a non-negative weight for each training input quantifying its resemblance to the new input, where such weights are specified by a predetermined function of training data vectors and the new vector, and which does not involve any optimization or tunable parameters;generating a sequence element following the starting fragment by combining the training outputs and their associated weights;associating with the generated element, the set of weights for individual training data fragments used to generate the said element, and defining the fractional provenance of the generated element in terms of the individual training data fragments to be given by the said weights;generating a new starting fragment by dropping the earliest element of the previous starting fragment, and appending the newly generated element from the previous step; and iterating the process of generating subsequent elements.

60. The method of claim 59, wherein the prediction process is statistically consistent.

61. The method of claim 59, wherein the weight associated with a training data input is proportional to an inverse power of the distance between the new input vector and a training input vector.

62. The method of claim 59, wherein the training sequences are composed of audio samples, or images, or video frames, or text (e.g., word tokens) suitable for language modeling, or financial information, or medical information.

63. The method of claim 59, wherein training data provenance for the generated sequence is computed by adding up all the weights created during the generative process associated with individual training data elements and using this sum to assign proportionate provenance corresponding to each training data element or subsets of training data elements.

64. The method of claim 63, wherein the power is given by the dimension of the input vectors.

65. A computer implemented method for sequential data compression, comprising:receiving an input sequence;fixing a subsequence length T, which length is fixed in advance, or user selected for the sequence in hand;creating a set of time-ordered training input-output pairs, where the training inputs are frames consisting of consecutive elements of the input sequence of fixed length T, and the training output for each frame is the sequence element immediately following the said fragment;storing the first frame and the sequence element immediately following that frame in a data store for holding the compressed sequence;shifting the frame forward by one time point, and predicting the next time point using the method of claim 48, with the training data input-output pairs consisting of frames preceding the current frame and their immediately following sequence elements;computing the residual between the predicted sequence element following the current frame, and the actual sequence element, and storing that residual in the data store for holding the compressed data; andrepeating the above two steps until the input sequence is exhausted.

66. The method of claim 65, wherein the sequence to be compressed is a sequence of audio samples, or a sequence of image fragments, or a sequence of video samples, or a sequence of text (e.g., word tokens), or a sequence of financial information, or a sequence of medical information, or any other signal that would benefit from data compression.

67. The method of claim 65, wherein the residuals are further compressed using quantization of the values or another means.

68. The method of claim 67, wherein the sequence to be compressed is a sequence of audio samples, or a sequence of video samples, or a sequence of word tokens, or any other signal that would benefit from data compression.

69. The method of claim 65, wherein the prediction process is statistically consistent.

70. The method of claim 65, wherein the weight associated with a training data input is proportional to an inverse power of the distance between the new input vector and a training input vector.

71. The method of claim 65, wherein the power is given by the dimension of the input vectors.

72. A computer implemented method for decompressing the stored compressed data generated by the procedure in claim 65, comprising:receiving a compressed data store generated by the procedure in claim 65; initiating the decompressed sequence using the stored first frame and the stored sequence element following that frame;for each subsequent sequence element to be appended to the decompressed, computing the predicted sequence element using the decompressed sequence up to the current position, as the source of training input-output pairs;retrieving the stored residual for the predicted sequence element, and adding that residual to the predicted sequence element to obtain the decompressed sequence element; and repeating the two above steps until the compressed data store is exhausted.

73. The method of claim 72, wherein the sequence to be compressed is a sequence of audio samples, or a sequence of image fragments, or a sequence of video samples, or a sequence of text (e.g., word tokens), or a sequence of financial information, or a sequence of medical information, or any other signal that would benefit from data compression.

74. The method of claim 72, wherein the prediction process is statistically consistent.

75. The method of claim 72, wherein the weight associated with a training data input is proportional to an inverse power of the distance between the new input vector and a training input vector.

76. The method of claim 72, wherein the power is given by the dimension of the input vectors.

77. A model or network free method for feedback control comprisingreceiving an initial pair of training sequences of measurements constituting system outputs, the corresponding sequence of controlling inputs, and a cost function that takes one or more values of the measurements and the values of the corresponding controlling inputs, and produces a positive cost as an output;using the method of claim 48, with the sequence of past measurements, and control values as inputs and the measurements at subsequent time points as outputs, to predict the system output at the next time step for a hypothetical control input at the present time, and computing the predicted resulting cost using the predicted output and the hypothetical control input;iteratively repeating the preceding step of predicting output for hypothetical control inputs, for a fixed time horizon into the future comprising one or more time steps, and computing and adding the corresponding predicted costs to obtain the total predicted cost for that sequence of hypothetical control inputs;repeating the procedure in the previous two steps with different choices of the sequences of hypothetical control inputs, and subsequently choosing that optimal sequence of hypothetical control inputs in which minimizes the total predicted cost;executing the first control input of that optimal sequence, and receiving the subsequent output measurement; andadding the new control input and the subsequent measured output to the training data sequences, and iterating the process described in the previous steps.

78. A model or network free method for reinforcement learning, comprising receiving an initial pair of training sequences of measurements constituting system outputs, the corresponding sequence of controlling inputs, and corresponding reward values;using the method of claim 48, with the sequence of past measurements, rewards and control values as inputs and the measurement and rewards at subsequent time points as outputs, to predict the system output as well as the predicted reward value at the next time step for a hypothetical control input at the present time;iteratively repeating the preceding step of predicting outputs and rewards for hypothetical control inputs, for a fixed time horizon into the future comprising one or more time steps, and adding the corresponding rewards to obtain the total predicted reward for that sequence of hypothetical control inputs;repeating the procedure in the previous two steps with different choices of the sequences of hypothetical control inputs, and subsequently choosing that optimal sequence of hypothetical control inputs in which minimizes the total predicted reward;executing the first control input of that optimal sequence, and receiving the subsequent output measurement as well as reward value; andadding the new control input and the subsequent measured output and reward value to the training data sequences; anditerating the process described in the previous steps.

79. The method of claim 1, wherein:the training items include training input and output pairs, for which, along with the new input, an output prediction is desired together with a quality assessment, as well as a numerical measure of quality or reliability associated with each training item pair;computing the desired prediction by combining the training outputs and their associated weights; andcomputing the quality or reliability associated with the prediction by combining the training values and their associated weights.

80. The method of claim 1, wherein:the training items include training input and output pairs, for which, along with the new input, an output prediction is desired after removal of designated training data items, without a parameter optimization step;computing the weights after removing the designated training data items; and computing the modified prediction by combining the training outputs and their associated weights in the remaining training data set.

81. The method of claim 80, wherein each training data item is associated with a quality or reliability score, and the subset of training data items to be removed from the prediction is based on those quality scores.

82. The method of claim 1, wherein:the training items include training input and output pairs, for which, along with the new input, an output prediction is desired after addition of a second set of training data items without a parameter optimization step;computing the weights using an augmented training data set after addition of the second set of training data to the first set of training data; andcomputing the desired prediction by combining the complete set of training outputs and their associated weights.

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