Methods for capturing and processing sensor data

Abstract

A method for measuring and processing sensor data from spatially distributed sensors, comprising the following steps: a) Acquisition of sensor data by a large number of spatially distributed measurement agents (12a - 12p), each having at least one sensor (14a - 14p), b) Creating or optimizing a local model of a physical relationship based on the sensor data (14a - 14p) acquired by each measurement agent (12a - 12p), wherein each measurement agent (12a - 12p) creates its own local model, c) Transmitting information concerning the created local model or transmitting the entire model from each measurement agent (12a - 12p) to its neighboring measurement agents (12a - 12b) that are within its transmission range, d) Reacquisition of sensor data (14a - 14p) by the multitude of spatially distributed measurement agents (12a - 12p), e) Iterative repetition of the procedure steps b) to d), wherein in procedure step b) the information received in the previous iteration from all neighboring measuring agents (12a - 12p) of a measuring agent is taken into account by this when creating its local model of the physical relationship, f) wherein the iteration according to procedure step e) is terminated when the local model of all measurement agents (12a - 12p) is identical, so that a global model is created, g) wherein the creation or optimization of the local model according to process step b) is carried out using a semismooth Newton method, characterized in that the local model is created or optimized on each measurement agent and subsequently the information concerning the optimized local model is transmitted to the immediately neighboring measurement agents, whereby this occurs as soon as new sensor data are available on a measurement agent, without having to wait for the complete filling of a data buffer for the sensor data, thus enabling optimized processing of time-variant processes.

Description

[0001] The invention relates to a method for acquiring and processing sensor data from spatially distributed sensors.

[0002] The spatially distributed sensors can be integrated into a variety of mobile measurement agents, such as mobile robots, intelligent wireless sensors, swarms of satellites, etc. Each of these measurement agents can communicate with its neighboring agents and is also capable of acquiring and internally processing sensor data. The large number of spatially distributed measurement agents allows sensor data to be collected over a larger area. For example, a swarm of robots can measure the magnetic field on a planet using sensor data.

[0003] The data collected by the measurement agents therefore differs depending on the location of the respective measurement agent.

[0004] For example, to estimate the distribution of the magnetic field on a planet, the sensor data collected by the measuring agents can be transmitted to Earth. Alternatively, they can be transmitted to a central computing unit on the planet, which will then perform a central estimate of the magnetic field and communicate the results back to the robots.

[0005] Frequent data communication should be avoided, however, as the necessary bandwidth is often unavailable, or even if it is, such frequent data communication consumes a lot of energy that is then unavailable for other tasks. It is therefore advantageous to process the collected sensor data locally at the respective measuring device, instead of evaluating the data centrally.

[0006] Informationen zum Stand der Technik können den folgenden Veröffentlichungen entnommen werden: [1] R. Olfati-Saber, J. Fax, and R. Murray, „Consensus and cooperation in networked multi-agent systems,“ Proc. IEEE, vol. 95, no. 1, pp. 215-233, Jan 2007. [2] G. Mateos, J. A. Bazerque, and G. B. Giannakis, „Distributed Sparse Linear Regression,“ IEEE Trans. Signal Process., vol. 58, no. 10, pp. 5262- 5276, Oct. 2010. [3] R. Tibshirani, „Regression shrinkage and selection via the LASSO,“ J. Roy. Statist. Soc., vol. 58, pp. 267-288, 1994. [4] S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, „Distributed optimization and statistical learning via the Alternating Direction Method of Multipliers,“ Foundations and Trends in Machine Learning, vol. 3, no. 1, pp. 1-122, jan 2011. [5] S.-J. Kim, K. Koh, M. Lustig, S. Boyd, and D. Gorinevsky, „An interior-point method for large-scale 1-regularized least squares,“ IEEE J. Selected Topics in Sig. Process., vol. 1, no. 4, pp. 606-617, Dec 2007. [6] M. Figueiredo, R. Nowak, and S. Wright, „Gradient projection for sparse reconstruction: Application to compressed sensing and other inverse problems,“ IEEE J. Selected Topics in Sig. Process, vol. 1, no. 4, pp. 586-597, Dec 2007. [7] R. Griesse and D. Lorenz, „A semismooth Newton method for Tikhonov functionals with sparsity constraints,“ Inverse Problems, vol. 24, no. 3, mar 2008. [8] A. Theodore, K. Oliver, L. Dirk, S. Stefan, and S. Klaus, „An active set approach to the elastic-net and its applications in mass spectrometry,“ in Signal Processing with Adaptive Sparse Structured Representations (SPARS'09), vol. 1, Saint Malo, France, 2009, pp. 1-4. [9] K. Bredies and D. Lorenz, „Linear convergence of iterative soft-thresholding,“ Journal of Fourier Analysis and Applications, vol. 14, no. 5-6, pp. 813-837, 2008.

[10] E. Eksioglu, „Rls adaptive filtering with sparsity regularization,“ in In Proc. of 10th Int. Conf. on Inform. Sciences Sig. Process. and Applic., May 2010, pp. 550-553.

[11] I. Daubechies, M. Defrise, and C. D. Mol, „An iterative thresholding algorithm for linear inverse problems with a sparsity constraint,“ Communications in Pure and Applied Mathematics, vol. 57, no. 11, pp. 1413-1457, 2004.

[12] A. Beck and M. Teboulle, „A fast iterative shrinkage-thresholding algorithm for linear inverse problem,“ SIAM J. on Imaging Sciences, vol. 2, no. 1, pp. 183-202, 2009.

[13] N. Parikh and S. Boyd, „Proximal algorithms,“ Foundations and Trends in Optimization, vol. 1, no. 3, pp. 123-231, 2013.

[14] X. Chen, Z. Nashed, and L. Qi, „Smoothing methods and semismooth methods for nondifferentiable operator equations,“ SIAM Journal on Numerical Analysis, vol. 38, no. 4, pp. 1200-1216, 2000

[15] G. H. Golub and C. F. Van Loan, Matrix Computations, 3rd ed. Baltimore, MD: The Johns Hopkins University Press, October 1996

[0007] It is known from the prior art that the local model of a physical relationship created by the measurement agents is generated using a sparse linear regression procedure, resulting in a linear model that can be described by a small number of non-zero parameters. This use of a smaller number of non-zero parameters ("sparsity") means that fewer unknown parameters need to be estimated, which in turn reduces the number of measurements required by the sensors. This leads to lower energy consumption, allowing for more efficient planning of the measurement agent missions. An efficient strategy for solving a sparse estimation problem in a distributed system was described in publication [2].

[0008] A disadvantage of the state of the art is that such a sparsely populated model can only be determined by using certain penalty functions that must be inserted into the optimization problem. These penalty functions are not smooth ("non-smooth penalties") and therefore pose particular challenges for distributed optimization of the model.

[0009] An effective strategy for estimating a sparse estimation problem in a distributed system consisting of multiple intelligent measurement agents was described in publication [2] for LASSO-type sparse solvers. Instead of a global (centralized) solution, a local solution was sought for each measurement agent, subject to the constraint that the local estimate of a measurement agent is identical to the estimates of its respective neighbors. The corresponding LASSO problem is then solved using a technique known as the alternating directions method of multipliers (ADMM), which is described in publication [2].The global optimization of the model is divided into two interrelated loops: an outer loop that attempts to find a common model (global consensus) among the measurement agents, and an inner loop that solves a local augmented LASSO problem. In publication [2], the inner loop is solved by using additional auxiliary variables within the ADMM. This simplifies the optimization for agents that, for example, cannot provide large energy reserves. However, the rate at which the local models of each measurement agent converge to a common global model slows down due to the required switching between the additional auxiliary parameters and the model parameters.

[0010] Other methods for solving a local LASSO problem are described, for example, in publications [5] and [6]. These methods use gradient information and can also be used to solve the optimization problem in the inner loop.

[0011] Possibilities for solving distributed optimization problems in the context of sensor networks are described in the following publication: MOTA, JF de C.: Communication Efficient Algorithms for Distributed Optimization. PhD Thesis, Universidade Tecnica de Lisboa, Instituto Superior Tecnico and Carnegie Mellon University, October 2013, pp. i-xviii and 1-120.

[0012] Further relevant state of the art is known from the following publication: HOUSKA, B. [et al]: An augmented Lagrangian based algorithm for distributed nonconvex optimization. Available online on September 22, 2015 at the URL: https: / / web.archive.org / web / 20150922044056 / http: / / www.optimizationonline.org / DB_FILE / 2014 / 07 / 4427.pdf

[0013] The object of the invention is to provide a method for acquiring and processing sensor data from spatially distributed sensors, which enables faster creation of a common global model of a physical relationship.

[0014] The problem is solved according to the invention by the features of claim 1.

[0015] Gradient-based optimizations are generally inferior to Newton-like optimizations, which take into account information about the curvature of the function (i.e., its second derivative). Unfortunately, this second derivative is not available for non-smooth functions, i.e., those necessary for estimating sparse model parameters. One technique that can be used to accelerate local optimization is the semismooth Newton (SSN) method. This is described, for example, in Publication 7. SSN methods allow for a very efficient solution of the local optimization problem.

[0016] Furthermore, adaptive LASSO-type algorithms are known from the prior art (see Publication 10). However, these methods are not applicable to distributed agent networks.

[0017] A disadvantage of gradient-based methods is their very slow speed. In a distributed system, this leads to slower convergence of the local model created by each measurement agent into a common global model. This speed can be increased by using Newton methods. However, these are not applicable to non-smooth functions.

[0018] By using the semismooth Newton method to create or optimize the local model, in this case the measurement agent, the time it takes for the local models of each measurement agent to converge is reduced, resulting in a common global model.

[0019] According to the invention, it is possible to limit the number of iterative repetitions according to process step e to a maximum of 5 and, in particular, a maximum of 3. Furthermore, it is possible to perform only one iteration.

[0020] It is preferred that the local model created by the measurement agents is created using a sparse linear regression procedure, resulting in a linear model that can be described by a small number of non-0 parameters.

[0021] Furthermore, it is preferred that the measurement agents are mobile and communicate wirelessly with each other.

[0022] The inventive method accelerates the lasso-based optimization in the inner loop by means of a Newton-like optimization. Applying the semismooth Newton method to non-smooth functions allows the evaluation of a set similar to a Hessian matrix in a standard Newton update. Thus, optimizations can be performed more efficiently than in gradient-based methods. This speeds up the inner loop.

[0023] A disadvantage of the prior art semismooth Newton method is that it is unstable under normal conditions and therefore requires further regulation to function stably. It is necessary that each subset of the measurement-related columns be injective into the matrix in a regression model. This requirement does not need to be met in the method according to the invention. On the contrary, investigations by the applicant have shown that the semismooth Newton method in the invention stabilizes itself through the distributed acquisition and processing of the sensor data for optimizing the respective local model. This leads to an acceleration of the model creation in the inner loop, which in turn accelerates the outer loop, namely the achievement of the entire network consensus.Furthermore, the application is enabled in an adaptive distributed system, thus enabling the processing of data lifestreams and simplifying the requirements regarding data storage at the individual measurement agents.

[0024] According to the invention, a local update of the local model of each measurement agent is performed within an inner loop. Within the outer loop, this updated model is then communicated to the immediately adjacent measurement agents. The invention allows the number of iterations of the inner and outer loops to be decoupled. For example, it is possible to perform several iterations in the inner loop before the outer loop is executed. Conversely, the number of iterations in the outer loop can also be increased without increasing the number of iterations in the inner loop.

[0025] The method according to the invention makes it possible to execute the inner and outer loops as soon as sensor data is available in a measurement agent. It is therefore not necessary to wait until a data buffer is filled in order to update the local model and communicate with neighboring measurement agents. This is the case in publication [2], however, where data processing is performed in blocks. This is disadvantageous for time-varying processes because changes in the physical context can occur during the waiting time until the data buffer is full, and these changes are not taken into account. According to the invention, such time-varying processes can be handled more effectively.

[0026] A specific embodiment of the method according to the invention is explained below with reference to figures: Fig. Figure 1 shows a schematic representation of a multitude of measurement agents 12a to 12p that acquire local sensor data using a multitude of sensors 14a to 14p. This sensor data is used to generate a local model of a physical relationship. This physical relationship can be any relationship that describes a situation in the real world (e.g., the physical model of a magnetic field or other physical relationships). Fig. Figure 2 shows estimated MSE results as a function of consensus iterations for different values ​​of c. Fig. Figure 3 shows different MSEs as a function of consensus iterations.

[0027] More specific features of the method according to the invention are described in more detail below: This work investigates the application of a technique known as the Semismooth Newton (SSN) method to accelerate the convergence of a distributed quadratic program LASSO (DQP-LASSO)—a consensus-based distributed sparse linear regression algorithm. The DQP-LASSO algorithm uses an alternating directions method of multipliers (ADMM) to reduce a global LASSO problem to a series of local (per-agent) LASSO optimizations, which are then distributed among neighbors in a network. The SSN algorithm has the advantage of superlinear convergence, thus enabling a more efficient implementation of these local optimizations. However, the SNN might encounter convergence problems in some cases. Here, it is shown that the ADMM-inherent regularization also provides sufficient regularization to stabilize the SSN algorithm, ensuring stable convergence of the entire scheme.Furthermore, the structure of the SSN algorithm also enables an adaptive implementation of distributed sparse regression. This allows for the estimation of time-varying sparse vectors and has a beneficial impact on the storage requirements for processing data streams. 1 Introduction

[0028] In this paper, we address a problem of decentralized distributed sparse linear regression over a network of interconnected intelligent agents: mobile robots or smart wireless sensors. Decentralized processing within the network relies on successive refinements of the agent's local model estimates. Such a distributed scheme typically comprises two steps: (i) an exchange of intermediate processing results between neighbors, and (ii) a local update performed by each agent. In cases where agents and their communication links are equivalent, it is reasonable to expect that local (per-agent) results should eventually agree across the network. Algorithms that achieve this are generally called consensus algorithms [1]. Here, we will discuss a specific type of consensus-based algorithm for distributed sparse linear regression. In sparse linear regression, we generally assume that each agent's measurements are represented by a linear model that has some non-zero parameters; this is common to all agents in the network, thus providing an incentive to use schemes adapted to the consensus-type model.However, the sparsity assumption introduces an additional constraint in the form of a loss of smoothness into the optimization problem, which poses particular challenges for distributed optimization.

[0029] An effective strategy for solving a sparse estimation problem in a distributed environment was proposed in [2] for LASSO-type functionals [3]. In [2], the authors formulate a constrained optimization problem to enforce consensus by requiring that individual agents' estimates match the estimates of the agent's neighbors. The corresponding LASSO problem is then solved using a technique known as the alternating directions method of multipliers (ADMM) [4]. This decouples global optimization into two optimization loops: an outer loop where consensus is enforced between agents, and an inner loop that solves a local, extended LASSO problem. Finding an efficient solution for this inner-loop optimization is the motivation for the present work.

[0030] In [2], this local LASSO update was achieved by using additional auxiliary variables in the ADMM. This simplifies the optimization, e.g., for performance-dependent agents, although convergence is slowed due to the additional alternation between auxiliary and model parameters. Of course, a number of other techniques can be used to solve the local LASSO problem, e.g., [5, 6], to name a few. However, most of these methods primarily use gradient information about the target. Here, we propose using a technique that implements a Newton-type optimization instead. This technique is known as Semismooth-Newton (SSN) [7] and is based on the concept of slant or generalized differentiability.When applied to non-smooth functionals, it allows the evaluation of a quantity similar to a Jacobian matrix at a standard Newton update, thus performing optimizations more efficiently compared to gradient methods.

[0031] Nevertheless, a disadvantage of the SSN technique is that it requires additional regularization to operate stably [8]; SSN requires any subset of measurement matrix columns in a regression model to be injective [9, 7]. We will demonstrate here that in a distributed environment, ADMM-based consensus alone provides sufficient regulation to stabilize the SSN procedure. The SSN technique not only accelerates inside-loop convergence but also results in faster network consensus, as demonstrated by test results. Furthermore, the structure of the SSN updates can be used to implement the estimation scheme adaptively. Although adaptive LASSO-type algorithms have been proposed (e.g.,

[10] ), these are also gradient-based and have not been studied in distributed environments.

[0032] Throughout this entire treatise, we will use the following system of notation. Calligraphic letters will be used. A used to denote the index sets. [X]A denotes a subset of columns of X, where indices in A are specified; described in a similar way [x]A a vector, where elements of x have indices in A are specified, and [x] l is the lte element of x. We further write [B]AJ=[X]AT[X]J, to denote the corresponding correlation matrix. 2 Signal Model

[0033] Assume a network of N interconnected agents. The network is represented as an undirected graph. G(N,ε) modeled, with the vertices N={1,…,N} correspond to the agents, and the edges ℇ represent links between connected agents in the network. Each agent n∈N can only interact with his immediate neighbors Nn⊂N communicate. Let us assume that the graph G(N,ε) connected in the sense that a (possibly multi-hop) route exists between each pair of agents.

[0034] Each agent leads M n linear observations of a K-sparse, L-dimensional signal w ∈ ℝ L according to the model yn=Xnw+ξn, through, where y n ∈ ℝ Mn The agent's measurement vector is X n ∈ ℝ Mn×L the measurement matrix is, and ξ n ∈ ℝ Mn a random perturbation vector. Furthermore, we will assume that ξn,n∈N, all independent, normal random variables with zero expected value and homoscedastic covariance matrix σξ2I are.

[0035] The purpose of the network is to obtain a common sparse estimate w by solving w^=argminw12∑n∈N‖yn−Xnw‖22+σξ2λ‖w‖1, which is also known as the LASSO regression problem

[15] . 2.1 ADMM technique for problems of distributed LASSO

[0036] A distributed solution to (2) using the ADMM method was proposed in

[11] . The main idea behind the method is (i) to introduce N local variables w n , and (ii) in the requirement that w1 = w1 = ... = w N , i.e., that individual estimates agree across the network. This converts (2) into a constrained optimization problem with separable objective functions: {w^n}n∈N=argmin{wn}n∈N∑n∈N12‖yn−Xnwn‖22+λN‖wn‖1,st wn=wn,n∈N,n'∈Nn. where we now σξ2 The application of the ADMM technique to solve (3) leads to a distributed optimization scheme summarized in Algorithm 1 (see also [2] for further details). A regularization constant c > 0 is an ADMM penalty parameter of the enlarged Lagrange function for problem (3) [2]. Algorithm 1, expressed in [2] as distributed quadratic program LASSO (DQP-LASSO), solves (2) by means of a series of local LASSO updates, which are then averaged among neighboring agents to reach consensus. 1 1 These consensus updates are represented in algorithm 1 with a superscript index i. Equation (5) obviously represents the largest computational load on the agent. An efficient solution to (5) can improve the performance of the entire network. To achieve this, we propose the use of a technique known as the Semismooth Newton (SSN) method [7]. As we will show, the SNN method not only accelerates the convergence of DQP-LASSO but also allows for an adaptive implementation of the distributed estimation scheme.

[0037] Below, we summarize the essential steps of the SSN algorithm within a DQP-LASSO framework. For further information on SSN, the reader is referred to [7]. 3 The Semismooth Newton Method

[0038] In [2] the authors have shown that by defining y˜n[i]=(yn12c|Nn|(c∑n'∈Nn(wn[i]+wn'[i])−pn[i])), and X˜n=(Xn2c|Nn|I), The optimization (5) can be written as a standard LASSO problem: w^n[i]=argminw12‖y˜n[i]−X˜nw‖22+λN‖w‖1.

[0039] A simple method for solving (7) is iterative soft thresholding (ISTA) or fast ISTA (FISTA) [11, 12], which are special forms of a proximal gradient method

[13] . ISTA is characterized by solving the following equation: f(w)=w−δηλ(w−ηX˜nT(X˜nw−y˜n[i]))=0,(8) where δα(x)=max{0,|x|−α}sign(x) a soft thresholding operator is and η=1 / ‖X˜nTX˜n‖ a (fixed) step size of the gradient update (for further details see [3, 13]).

[0040] Nevertheless, proximal gradient methods or similar techniques that only use the “gradient” information about the target converge at a linear rate [9]. In contrast, the SSR algorithm converges superlinearly. It is based on the concept of a generalized or skewed differentiability of a soft-thresholding operator, which allows a more efficient solution of (8)

[14] . In particular, SSN solves (7) iteratively by computing w^n[i,m+1]=w^n[i,m]−G−1(w^n[i,m])f(w^n[i,m]), where the superscript [i,m] denotes the m-th SSN iteration on the i-consensus iteration and G(w^n[i,m]) a generalized derivative of f(w), evaluated at w^n[i,m], Roughly speaking, (9) can be understood as a Newton update, where the unavailable Hessian matrix of (7) is replaced by a generalized derivative.

[0041] Following [7] G(w^n[i,m]) calculated as follows. First, we define for L={1,…,L} an active set A∈L and an inactive set J∈L: εn[i,m]=w^n[i,m]−ηΦ˜nw^n[i,m]+ηψ˜n[i],A={l∈L:|[εn[i,m]]l|>ηλN,},J=L\A, where Φ˜n=X˜nTX˜n and ψ˜n[i]=X˜nTy˜n[i]. The generalized derivation of (8) at w^n[i,m] is then calculated as [7]: G(w^n[i,m])=(η[Φ˜n]AAη[Φ˜n]AJ0I), where [Φ˜n]AJ=[X˜n]AT[X˜n]J. The pseudocode of the SSN scheme is summarized in Algorithm 3.

[0042] It should be noted that the superlinear convergence of the SSN algorithm depends on the boundedness of G. -1 (w) is based [7]. The latter property is satisfied under the condition that X̃ n satisfies a so-called finite basis injectivity (FBI) property. FBI states that every subset of columns of X̃ nis injective. In general, the absence of the FBI property could impair the convergence of the SSN scheme. Surprisingly, however, when SSN is used in the DQP-LASSO environment, the constraints of G -1 (w) guaranteed, as summarized in the following proposal.

[0043] Proposal 1 In a connected network, it is guaranteed that a linear mapping X̃ defined in (6) n has a finite basis injectivity property, i.e., for all subsets A⊂{1,…,N} is the image [X˜n]A injective, and G -1 (w) is restricted.

[0044] The FBI essentially guarantees that the Matrix [Φ˜n]AA is invertible and therefore G -1 (ŵ n ) exists. Let it be a subset A⊂{1,…,N} assumed, and it be [I]A the corresponding subset of columns of the L × L identity matrix I. Obviously, the following holds: [Φn]AA=[I]ATX˜nTX˜n[I]A=[I]AT(XnTXn+(2c|Nn|)I)[I]A=[Xn]AT[Xn]A+(2c|Nn|)I for a connected network |Nn|>0; Thus, the ADMM regularization constant c > 0, which reflects the compromise between the consensus conditions and the model fit in the enlarged ADMM Lagrange function, also ensures that [Φ˜n]AA has a full rank. The latter implies the FBI attribute for X̃. n .

[0045] It should also be noted that calculating G -1 (ŵ [i,m] ) the inversion of a |A|×|A|−Matrix[Φ˜n]AA This inverse can be efficiently computed if elements are added to the active set during SSN iterations. A added or removed. Especially if the active set A Changing the m-th SSN iteration corresponds to inserting or removing columns in the matrix. [X˜n]A. The corresponding update from ([X˜n]AT[X˜n]A)−1 can then be effectively calculated using an existing inverse. [Φ˜n]AA−1, which is available at the (m - 1)th iteration, and standard results for block Gaussian elimination

[15] . 4 Adaptive distributed linear regression

[0046] SSN iterations not only accelerate the convergence of local DQP-LASSO updates, but also allow for an adaptive implementation of the algorithm, in which data is processed sequentially. This reduces the storage requirements on each agent and enables the processing of live data streams.

[0047] Let us now assume a measurement model (1) of the following form: yn,j=un,jTwj+ξn,j where y n,j ≡ [y n,0 , ... ,y n,j ] T , and X n,j ≡ [u n,0 , ..., u n,j ] T Here is w j Now, a function of j. Our intention here is to show how an active set set A and [Φ˜n]AA−1 can be calculated recursively for the model (13).

[0048] We now define a matrix Φ̃ n,j as follows: Φ˜n,j=∑l=0jμj−lun,lun,lT+2c|Nn|I=μΦn,(j−1)+un,jun,jT+2c|Nn|I. where we have used (6) and introduced a forgetting factor 0 << µ < 1, which results in an exponential weighting of older measurements. Furthermore, we define ψ˜n,j[i]=∑l=0jμj−lun,lyn,l+αn,j[i]=μψn,(j−1)+un,jyn,j+αn,j[i], where αn,j[i]=(c∑n'∈Nn(w^n,j[i]+w^n',j[i])−pn,j[i]), and pn,j[i] is given in (4). Both Φ̃ n,j as well as ψ˜n,j[i] are updated recursively when a new measurement is taken or when new consensus information is received. The corresponding active and inactive sets are then calculated as in (10), with substitutions. ψ˜n[i]=ψ˜n,j[i] and Φ˜n=Φ˜n,j

[0049] The inverse [Φ˜n,j]AA−1 can also be calculated efficiently. It should be noted that when a new measurement is received, [Φ˜n,j]AA−1 before updating the active set A is updated, i.e., before the new SSN iterations. 4.0.1 The case µ = 1

[0050] In this case, [Φ˜n,j]AA−1 recursively calculated from [Φ˜n,(j−1)]AA−1 using the ratio [Φ˜n,j]AA−1=([I]AT(Φ˜n,(j−1)+un,jun,jT)[I]A)−1=([Φ˜n,(j−1)]AA+[un,j]A[un,j]AT)−1,

[0051] The latter inversion is a simple rank one update of [Φ˜n,(j−1)]AA−1. 4.0.2 The case µ < 1

[0052] In this case, the calculation becomes somewhat more complicated, as it requires a direct evaluation of the inversion, i.e. [Φ˜n,j]AA−1=([I]ATΦ˜n,j[I]A)−1.

[0053] It should be noted that this inversion is only calculated when a new measurement is included in the model.

[0054] 4.0.3 Updating from [Φ˜n,j]AA−1 during consensus and SSN iterations

[0055] During consensus iterations i and SSN iterations m, the active set could vary, allowing for the introduction or removal of columns from X̃. n corresponds to the relevant update from [Φ˜n,j]AA−1 can then be efficiently implemented using standard results for block Gaussian elimination

[15] .

[0056] The corresponding optimization steps are summarized in Algorithm 3. 5 results

[0057] We use a simple, hypothetical example to demonstrate the performance of the scheme. For this purpose, we generate a network with N = 9 agents with a random topology and 3 to 4 neighbors per agent. Assume that the true vector w has L = 50 elements, where K = 15 elements are non-zero and placed at randomly determined, unknown positions. The amplitudes of the non-zero entries are randomly chosen to be ±1. The measurement matrices in (1) are also randomly generated, with elements taken from a standard normal distribution. The noise variance σξ2 The value is chosen such that it corresponds to a signal-to-noise ratio (SNR) of approximately 20 dB. The regularization parameter λ = 5 was selected using cross-validation.

[0058] The performance of the proposed DQP-SSN-LASSO algorithm was compared with a D-LASSO algorithm from [2] and with a centralized SSN algorithm. We will also evaluate the performance of the adaptive version of the DQP-SSN-LASSO algorithm (Ad-LASSO) with µ = 1. In all compared cases, the number of measurements per agent is set to M. n = 20 set. Ad-LASSO is further restricted in that it cannot exceed M n Measurements are performed to make the results comparable with other schemes. As a performance criterion, we calculate the mean squared error (MSE) between the estimated w^n[i] and a true vector w. The corresponding curves, averaged over 10 independent runs, are in Fig. 1 summarized.

[0059] Fig. Figure 2 shows estimated MSE results as a function of consensus iterations for different regularization values ​​c.

[0060] As expected, the DQP-SSN-LASSO algorithm outperforms the D-LASSO algorithm for various regularization values ​​c. Smaller regularization values ​​c require fewer iterations; however, it should be noted that a smaller c could negatively impact the stability of both the ADMM and SSN algorithms, particularly under low SNR conditions. Furthermore, as the number of consensus iterations increases, all algorithms approach a centralized solution provided by SSN. It is also worth mentioning that the number of internal SSN iterations can be kept quite low in practice without any degradation in performance. The simulations presented here use only three internal SSN iterations.

[0061] Finally, we consider an adaptive situation in which the vector w is regenerated after 100 consensus iterations. The corresponding convergence results, averaged over 10 arbitrary iterations, are presented in Fig. Figure 2 shows that with a forgetting factor µ < 1, the algorithm is able to follow changes in the model parameters. 6 Concluding Remarks

[0062] This paper investigates a technique for accelerating the convergence rate of a distributed solution to a LASSO problem using the ADMM optimization scheme. The acceleration is achieved by employing a semismooth Newton (SSN) algorithm to implement local optimization updates per agent more efficiently. While convergence problems can generally arise with the SSN technique, we have shown that the regularization inherent in the ADMM scheme proves sufficient to stabilize the SSN algorithm. We further demonstrate that the structure of the SSN updates can be used to implement the optimization adaptively. This enables recursive processing of the data and the estimation of time-varying sparse vectors.

[0063] Fig. Figure 3 shows the estimated MSE as a function of consensus iterations.

Claims

A method for measuring and processing sensor data from spatially distributed sensors, comprising the following steps: a) Acquiring sensor data by a plurality of spatially distributed measurement agents (12a - 12p), each having at least one sensor (14a - 14p), b) Creating or optimizing a local model of a physical relationship based on the sensor data (14a - 14p) acquired by each measurement agent (12a - 12p), wherein each measurement agent (12a - 12p) creates its own local model, c) Transmitting information concerning the created local model ord) Transmission of the entire model from each measuring agent (12a - 12p) to its neighboring measuring agents (12a - 12b) within its transmission range, e) Reacquisition of sensor data (14a - 14p) by the multitude of spatially distributed measuring agents (12a - 12p), e) Iterative repetition of steps b) to d), wherein in step b) the information received in the previous iteration from all neighboring measuring agents (12a - 12p) of a measuring agent is taken into account by that agent when constructing its local model of the physical relationship, f) wherein the iteration according to step e) is terminated when the local model of all measuring agents (12a - 12p) is identical, resulting in a global model, g) wherein the construction or optimization of the local model according to step b) is carried out using a semismooth Newton method This is characterized by the fact that the local model is created on each measurement agent.The optimized local model is then transmitted to the immediately neighboring measurement agents as soon as new sensor data is available at a measurement agent, without having to wait for a data buffer to be fully filled, thus enabling optimized processing of time variant processes. Method according to claim 1, characterized in that the local model created by the measurement agents (12a - 12p) is created by a sparse linear regression method, such that a linear model is created that can be described by a small number of non-0 parameters. Method according to claim 1 or 2, characterized in that the measuring agents (12a - 12p) are mobile and communicate wirelessly with each other. Method according to claims 1 to 3, characterized in that the iterative repetition of process steps b) to d) according to process step e) is carried out a maximum of five times and preferably a maximum of three times.

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