Fast multi-dimensional partial fourier transform method and apparatus for supporting automatic hyper-parameter selection

Abstract

To provide a high-speed multi-dimensional partial Fourier transform method and device for maintaining accuracy while automatically selecting a hyper parameter to be used for the polynomial approximation of Fourier transform, and quickly calculating a part of a Fourier coefficient to multi-dimensional data by utilizing the polynomial approximation.SOLUTION: The fast multi-dimensional partial Fourier transform method performed by the fast multi-dimensional partial Fourier transform device includes a step of setting a plurality of hyperparameters to be used for partial Fourier transform on the basis of an allowable error for polynomial approximation and a constraint on the order of a polynomial, and a step of approximating and calculating multi-dimensional Fourier coefficients of partial Fourier transform for multi-dimensional data on the basis of the plurality of hyperparameters.SELECTED DRAWING: Figure 6

Description

[Technical Field]

[0001] The embodiments disclosed herein relate to a method and apparatus for fast multidimensional partial Fourier transform, and more particularly to a method and apparatus for fast multidimensional partial Fourier transform that utilizes polynomial approximation to quickly calculate some of the Fourier coefficients for multidimensional data while maintaining accuracy.

[0002] This research was conducted as a result of the "Time Series Multidimensional Data Mining Using Ultra-High Performance Irregular Tensor Analysis" project (NRF-2022R1A2C3007921) of the Ministry of Science and ICT and the National Research Foundation of Korea (NRF)'s Individual Basic Research Project.

[0003] This research was conducted as a result of the "(SW Starlab) Development of flexible and efficient model compression technology to support various applications and environments" project (IITP-2020-0-00894) of the SW Computing Industry Source Technology Development Project of the Ministry of Science and ICT and the Institute for Information and Communications Technology Planning (IITP).

[0004] This research was conducted as a result of the Ministry of Science and ICT and the Institute for Information and Communications Technology Planning (IITP)'s "Support for Graduate School of Artificial Intelligence (Seoul National University)" project (IITP-2021-0-01343) and "Research and Development of Artificial Intelligence Innovation Hub" project (IITP-2021-0-02068) as part of the ICT Broadcasting Innovation Talent Development Project. [Background technology]

[0005] The Fourier transform (FT) is a transformation that decomposes an input signal into various frequency components and is expressed as a sum of periodic functions.

[0006] The Discrete Fourier Transform (DFT), which applies the Fourier transform to digital signal processing, is a core algorithm used in various data mining tasks, including anomaly detection, latent pattern extraction, image processing, etc. The Discrete Fourier Transform calculates the discrete Fourier coefficients of the input data and uses these as the core characteristics of the data.

[0007] The main algorithm used to calculate the discrete Fourier coefficients is the Fast Fourier Transform (FFT), which uses an inefficient method of calculating all the Fourier coefficients, then selecting only the necessary coefficients and discarding the rest.

[0008] The Partial Fourier Transform (PFT), introduced to solve this inefficiency problem, is an algorithm that calculates only a portion of the discrete Fourier coefficients of data to efficiently analyze the characteristics of the data.

[0009] The existing partial Fourier transform has a limitation that it is only useful when the number of Fourier coefficients to be calculated is very small compared to the size of the input data. In particular, the existing partial Fourier transform does not take into account the input of multidimensional data, and does not consider how to optimize the calculation process for a portion of the Fourier coefficients for multidimensional data.

[0010] For reference, Patent Document 1 is an invention related to an efficient implementation method of multidimensional fast Fourier transform, Patent Document 2 is an invention related to a fast partial Fourier transform method, and Patent Document 3 is an invention related to a variable fast Fourier transform device. Here, Patent Documents 1 to 3 only disclose general content regarding fast Fourier transform operations, but do not provide a fast partial Fourier transform technology for multidimensional data. [Prior art documents] [Patent documents]

[0011] [Patent Document 1] Korean Patent Registration No. 10-0592753 [Patent Document 2] Korean Patent Registration No. 10-2514264 [Patent Document 3] Korean Patent Registration No. 10-1297085 [Non-patent literature]

[0012] [Non-Patent Document 1] Yong-chan Park et al., Fast Partial Fourier Transform, 28 Aug 2020 (https: / / arxiv.org / abs / 2008.12559) Summary of the Invention [Problem to be solved by the invention]

[0013] The embodiments disclosed in this specification aim to provide a method and apparatus for fast multidimensional partial Fourier transform that automatically selects hyperparameters used in polynomial approximation of Fourier transform and utilizes polynomial approximation to quickly calculate some of the Fourier coefficients for multidimensional data while maintaining accuracy.

[0014] Other objects and advantages of the present invention will become apparent from the following description and examples, and will be readily apparent from the following description and examples, which can be realized by the means and combinations set forth in the claims. [Means for solving the problem]

[0015] As a technical means for achieving the above-described technical object, a fast multidimensional partial Fourier transform method executed by a fast multidimensional partial Fourier transform device includes a step of setting a plurality of hyperparameters to be used for the partial Fourier transform based on a tolerance for polynomial approximation and constraints on the degree of the polynomial, and a step of approximating and calculating multidimensional Fourier coefficients of the partial Fourier transform for multidimensional data based on the plurality of hyperparameters.

[0016] According to another embodiment, a fast multidimensional partial Fourier transform device includes a control unit that sets a plurality of hyperparameters to be used for the partial Fourier transform based on a tolerance for polynomial approximation and constraints on the degree of the polynomial, and approximates and calculates multidimensional Fourier coefficients of the partial Fourier transform for multidimensional data based on the plurality of hyperparameters.

[0017] According to yet another embodiment, the recording medium is a computer-readable recording medium having recorded thereon a program for performing a fast multidimensional partial Fourier transform method.

[0018] According to yet another embodiment, the computer program is a computer program stored on a recording medium for being executed by a fast multidimensional partial Fourier transform device to perform a fast multidimensional partial Fourier transform method. [Effects of the Invention]

[0019] According to any one of the above-mentioned means for solving the problems, a method and apparatus for fast multidimensional partial Fourier transform can be provided, which can approximate trigonometric factors of multidimensional data by multivariate polynomial approximation and efficiently calculate some of the Fourier coefficients of multidimensional data by utilizing tensor transform and tensor product when processing the multidimensional data.

[0020] Furthermore, according to any one of the above-mentioned means for solving the problems, it is possible to provide a method and apparatus for fast multidimensional partial Fourier transform that enables explicit reconstruction of complex constraints using polynomial approximation and automatically selects optimal hyperparameters using unconstrained convex optimization even when the input and output sizes for multidimensional data processing are changed. The effects obtained by the disclosed embodiments are not limited to those mentioned above, and other effects not mentioned will be clearly understood by those skilled in the art to which the disclosed embodiments pertain from the following description. [Brief explanation of the drawings]

[0021] The accompanying drawings below illustrate preferred embodiments disclosed in this specification and serve to further understand the technical ideas disclosed in this specification, along with specific content for implementing the invention, and therefore the content disclosed in this specification should not be interpreted as being limited solely to the matters depicted in the drawings.

[0022] [Figure 1] FIG. 2 illustrates an example input image and a Fourier map for the input image. [Figure 2] 1A and 1B are diagrams illustrating how an existing partial Fourier transform and a fast multidimensional partial Fourier transform device according to an embodiment process two-dimensional inputs, respectively. [Figure 3] FIG. 1 is a block diagram illustrating the functional configuration of a fast multidimensional partial Fourier transform device according to an embodiment. [Figure 4] 1 is a flowchart illustrating the overall operation of a fast multidimensional partial Fourier transform device according to an embodiment. [Figure 5] FIG. 2 is a diagram illustrating a tensor transformed by a fast multidimensional partial Fourier transform device according to an embodiment. [Figure 6] 1 is a flowchart of a fast multidimensional partial Fourier transform method according to one embodiment. [Figure 7] 1 is a flowchart of a fast multidimensional partial Fourier transform method according to one embodiment. [Figure 8] 1 is a flowchart of a fast multidimensional partial Fourier transform method according to one embodiment. DETAILED DESCRIPTION OF THE INVENTION

[0023] Various embodiments will be described in detail below with reference to the accompanying drawings. The embodiments described below may be implemented in various modified forms. In order to more clearly describe the features of the embodiments, detailed descriptions of matters that are well known to those skilled in the art to which the following embodiments pertain will be omitted. In addition, parts of the drawings that are not relevant to the description of the embodiments will be omitted, and similar parts will be designated by similar reference numerals throughout the specification.

[0024] Throughout the specification, when a certain component is said to be "connected" to another component, this includes not only "directly connected" but also "connected via another component in between." Furthermore, when a certain component is said to "include" another component, this does not exclude the other component, but means that the other component may also be included, unless otherwise specified.

[0025] First, the terms used in this specification will be explained.

[0026] A "hyperparameter" is a parameter that sets the details necessary to calculate a polynomial approximation for approximating the multidimensional Fourier coefficients of a partial Fourier transform of multidimensional data.

[0027] A "tensor" is a data type that represents a multidimensional array. Tensor data can be used to represent and operate on a variety of data, including text, images, sound, and video, on a computer. There are various ways to represent tensors on a computer, and typical examples include a rank indicating the dimension, a shape indicating the number of elements corresponding to each dimension, and a data type related to the value. For example, zero-dimensional data can be classified as a zero-dimensional tensor or scalar, a one-dimensional data array as a one-dimensional tensor or vector, a two-dimensional data array as a two-dimensional tensor or matrix, and a three- or higher-dimensional data array as a multidimensional tensor or N-dimensional tensor. Multiple tensors can be operated on together for data processing.

[0028] Hereinafter, the embodiments will be described in detail with reference to the accompanying drawings.

[0029] FIG. 1 is a diagram illustrating an input image and a Fourier map for the input image.

[0030] The multiple Fourier maps 121, 122, and 123 shown in Figure 1 visualize the Fourier coefficients for multiple input images 111, 112, and 113, and are maps that represent the Fourier coefficients in log magnitude. In the multiple Fourier maps 121, 122, and 123, most of the Fourier coefficients are close to zero, except for the low-frequency portion around the center. This indicates that computational efficiency can be achieved by focusing only on the non-zero coefficient portion and omitting the calculation of unnecessary coefficients. Most real-world data, such as time series data, images, and videos, has a highly compressed representation in the frequency domain. It is necessary to efficiently calculate only a portion of the Fourier coefficients by leveraging the energy compaction properties of the data.

[0031] FIG. 2 illustrates how a conventional partial Fourier transform and a fast multidimensional partial Fourier transform device according to an embodiment process two-dimensional inputs.

[0032] Non - Patent Document 1 is an existing partial Fourier transform algorithm that uses polynomial approximation to reduce the time complexity to O(N + MlogM). Here, N is the size of the input, and M is the size of the output region.

[0033] The existing partial Fourier transform algorithm according to Non - Patent Document 1 has two drawbacks.

[0034] First, since Non - Patent Document 1 is specially designed for one - dimensional inputs, its effectiveness decreases when applied to multi - dimensional data. For example, assuming that, as shown in Figure 2, low - frequency coefficients 220 of size T×T are calculated for a two - dimensional input 210 of size S×S, Non - Patent Document 1 operates by applying a number of one - dimensional partial Fourier transforms to each dimension. So, it requires a cost of S·(S + TlogT)+T·(S + TlogT)~S

[0035] + STlogT. The high - speed multi - dimensional partial Fourier transform device according to an embodiment applies the partial Fourier transform to the entire input once. So, it requires a cost of S 2 + T 2 logT 2 ~S 2 + T 2 logT. Here, since T << S, the high - speed multi - dimensional partial Fourier transform device according to an embodiment has a significant computational gain in multi - dimensional data processing compared to the existing partial Fourier transform algorithm.

[0035] Second, Non-Patent Document 1 relies on manual hyperparameter search. Given an input of size N, the user must directly select an appropriate divisor for N to use the existing partial Fourier transform algorithm described in Non-Patent Document 1. Because the overall performance of existing partial Fourier transform algorithms varies significantly depending on the divisor value, it is important to select the optimal divisor each time the input or output size is changed. However, existing partial Fourier transform algorithms do not offer an option to automatically find the optimal value, so in the worst case, the user must go through trial and error with every divisor. In particular, if the input is multidimensional, the situation becomes even worse because the search space for hyperparameters grows exponentially with the dimension. A fast multidimensional partial Fourier transform device according to one embodiment can significantly reduce the cost of hyperparameter search by automatically finding the optimal divisor value using an algorithm based on convex optimization.

[0036] To overcome these problems, the fast multidimensional partial Fourier transform apparatus according to the present embodiment automatically selects hyperparameters and efficiently and accurately calculates some of the Fourier coefficients for multidimensional data based on the automatically selected hyperparameters. The fast multidimensional partial Fourier transform apparatus according to the present embodiment performs two main operations.

[0037] First, the fast multidimensional partial Fourier transform device according to this embodiment calculates partial Fourier coefficients using a pre-calculation technique based on multivariate polynomial approximation of trigonometric functions. The fast multidimensional partial Fourier transform device approximates the trigonometric factor set of the partial Fourier transform using multivariate polynomials. The fast multidimensional partial Fourier transform device reduces computational costs by decomposing the partial Fourier transform into smaller sub-blocks and approximating some trigonometric functions with Chebyshev polynomials.

[0038] The fast multidimensional partial Fourier transform device according to this embodiment efficiently calculates partial Fourier coefficients using a constellation matrix multiplication and fast Fourier transform algorithm optimized for multidimensional data types. The fast multidimensional partial Fourier transform device uses the Cooley-Tukey algorithm to find a set of smooth twiddle factors in a multidimensional partial Fourier transform. The twiddle factors are then approximated using multivariate polynomials. This significantly reduces time costs by decomposing the calculation of the partial Fourier coefficients into matrix multiplication and a multidimensional fast Fourier transform of small subblocks of the input.

[0039] The fast multidimensional partial Fourier transform device according to this embodiment has superior performance to the existing partial Fourier transform algorithm, and improves the processing speed by up to 7.6 times without reducing accuracy.

[0040] Second, the fast multidimensional partial Fourier transform device according to this embodiment automatically finds optimal hyperparameters and calculates the degree of the approximate polynomial. By automatically selecting hyperparameters using an algorithm based on convex optimization without manual hyperparameter search, the additional cost required for hyperparameter search is significantly reduced.

[0041] The optimal hyperparameters derived by the fast multidimensional partial Fourier transform device according to this embodiment are values ​​that minimize the time complexity of multidimensional partial Fourier transform operations. Although the constraint functions of the optimization problem cannot be expressed in an explicit form, the fast multidimensional partial Fourier transform device induces an explicit reconstruction of the constraint functions based on Chebyshev polynomial approximation, thereby approximating the complex constraint functions of the multidimensional partial Fourier transform to an unconstrained convex optimization problem. This allows for efficient search for optimal hyperparameters using numerical analysis such as Newton's method.

[0042] FIG. 3 is a block diagram illustrating the functional configuration of a fast multidimensional partial Fourier transform device according to an embodiment.

[0043] 3, a fast multidimensional partial Fourier transform device 300 according to an embodiment may include an input / output unit 310, a memory 320, a control unit 330, and a communication unit 340. Some components may be omitted as necessary.

[0044] The input / output unit 310 may include an input unit for receiving input from a user and an output unit for displaying information such as a result of execution of a task or a status of the fast multidimensional partial Fourier transform device 300. That is, the input / output unit 310 is configured to receive input data and output a result of processing the input data. The fast multidimensional partial Fourier transform device 300 according to the embodiment may receive a fast multidimensional partial Fourier transform request, etc., via the input / output unit 310.

[0045] The memory 320 is configured to store files and programs and may be configured with various types of memory. In particular, the memory 320 may store data and programs that enable the control unit 330 (described later) to perform calculations for fast multidimensional partial Fourier transforms according to the algorithms presented below.

[0046] The memory 320 can store a plurality of hyperparameters used in the fast multidimensional partial Fourier transform, and can store partial Fourier coefficients for the multidimensional data calculated based on the plurality of hyperparameters.

[0047] The control unit 330 includes at least one processor such as a CPU, a GPU, or the like, and can control the overall operation of the fast multidimensional partial Fourier transform device 300. That is, the control unit 330 can control other components included in the fast multidimensional partial Fourier transform device 300 to perform operations for the fast multidimensional partial Fourier transform. The control unit 330 can execute a program stored in the memory 320 to perform an operation for approximating multidimensional Fourier coefficients by the fast multidimensional partial Fourier transform according to an algorithm to be presented below.

[0048] The communication unit 340 may perform wired or wireless communication with other devices or networks. For example, the communication unit 340 may receive multiple hyperparameters and transmit multi-dimensional Fourier coefficients.

[0049] For this purpose, the communication unit 340 may include a communication module supporting at least one of various wired and wireless communication methods, and the communication module may be implemented in the form of a chipset. The mobile communication or wireless communication supported by the communication unit 340 may be, for example, an N-generation mobile communication protocol, WiFi (Wireless Fidelity), Wi-Fi Direct, Bluetooth, UWB (Ultra-Wide Band), or NFC (Near Field Communication).

[0050] The control unit 330 automatically searches for optimal hyperparameters for calculating multidimensional Fourier coefficients. The control unit 330 sets multiple hyperparameters used in the partial Fourier transform based on the tolerance for polynomial approximation and constraints on the degree of the polynomial.

[0051] The control unit 330 defines optimal hyperparameters as values ​​that minimize the time complexity of the algorithm. The constraint functions of such optimization problems cannot be expressed explicitly. To solve this problem, the control unit 330 reconstructs the constraint functions using Chebyshev approximation to derive an unconstrained convex optimization problem. This approach induces convexity in the objective function, allowing for efficient search for optimal hyperparameters using numerical interpretations such as Newton's method. After searching for an optimal solution that minimizes the objective function, the control unit 330 can automatically derive optimal hyperparameters by approximating the optimal hyperparameters using a specific function.

[0052] The control unit 330 efficiently calculates partial Fourier coefficients in the multidimensional partial Fourier transform. The control unit 330 approximates and calculates multidimensional Fourier coefficients of the partial Fourier transform for the multidimensional data based on a plurality of hyperparameters.

[0053] The control unit 330 modifies the Cooley-Tukey algorithm to find a set of trigonometric functions with minimal oscillations using multidimensional partial Fourier transforms. The trigonometric functions are then approximated using Chebyshev polynomials. This method significantly reduces the time cost by decomposing the calculation of the partial Fourier coefficients into matrix multiplications and multidimensional fast Fourier transforms for small subblocks of the input.

[0054] The control unit 330 divides the configuration phase and the computation phase into two phases to calculate multidimensional partial Fourier transform coefficients at high speed. In the configuration phase, the control unit 330 performs multivariate Chebyshev polynomial approximation for the trigonometric functions of the partial Fourier transform and performs sequential tensor products with the input sub-blocks. The control unit 330 also automatically searches for optimal hyperparameters and calculates the degree of the approximation polynomial. In the computation phase, the control unit 330 uses configuration matrix multiplication and fast Fourier transform algorithms optimized for multidimensional data types.

[0055] JPEG2026009798000002.jpg26170

[0056]

number

[0057] where n = (n1, ..., nD) and m = (m1, ..., mD) ∈ ZD are the indices of the input and output.

[0058] JPEG2026009798000004.jpg33150

[0059] The control unit 330 can decompose Equation 1 into Equation 2 using the Cooley-Tukey algorithm, which recursively divides the magnitude N into two equal parts and divides and conquers until a signal of length 2 is obtained.

[0060]

number

[0061] JPEG2026009798000006.jpg11150

[0062] JPEG2026009798000007.jpg19150

[0063]

number

[0064] JPEG2026009798000009.jpg23169

[0065] JPEG2026009798000010.jpg41150

[0066]

number

[0067] Such a polynomial has the properties of uniqueness and existence.

[0068] The control unit 330 uses the Chebyshev approximation algorithm to calculate the optimal polynomial approximation. Chebyshev polynomials are used because of their wide application in achieving optimal approximations with respect to uniform norms and their solid theoretical foundation, including their contribution to the derivation of error bounds. Other types of orthogonal polynomials can be applied to improve accuracy and efficiency when necessary.

[0069] The control unit 330 calculates ξ(ε, r) as an exponential function e when a tolerance ε>0 and a positive integer r are given. πix Define the range about the origin over which r can be approximated by a polynomial of degree smaller than r with approximation bound ε.

[0070]

number

[0071] JPEG2026009798000013.jpg14150

[0072] If a tolerance ε>0 is given, the control unit 330 calculates ξ(ε,r d )≧M d / P d A positive integer r that satisfies d You can search for.

[0073] JPEG2026009798000014.jpg36150

[0074] The control unit 330 can approximate Equation 3 to Equation 4.

[0075]

number

[0076] JPEG2026009798000016.jpg30142

[0077] JPEG2026009798000017.jpg15150

[0078]

number

[0079] To calculate mathematical formula 5, a total of D! parentheses (Parenthesizations) are required.

[0080] The control unit 330 can pre-compute the optimal brackets given (N, M, μ, ε) in the construction phase and bypass the bracket problem in the computation phase.

[0081] JPEG2026009798000019.jpg34150

[0082]

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[0083] JPEG2026009798000021.jpg18150

[0084]

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[0085] JPEG2026009798000023.jpg23150

[0086] The control unit 330 provides a convex optimization-based algorithm for selecting optimal hyperparameters for the fast multidimensional partial Fourier transform. The control unit 330 converts the fast multidimensional partial Fourier transform optimization problem (Problem 1) into an unconstrained convex optimization problem (Problem 2) by approximating the constraint functions.

[0087] The optimal hyperparameters minimize the time complexity of the fast multidimensional partial Fourier transform. The control unit 330 sets the time cost function. Since the construction phase only includes data-independent processes, we only consider the computation phase for the time cost. For simplicity, we use the following notation: N=Π for d=1, 2, ..., D. d N d , M=Π d M d , p=Π d p d , q=Π d q d , r=Π d r d is.

[0088] JPEG2026009798000024.jpg18150

[0089]

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[0090] JPEG2026009798000026.jpg23150

[0091]

number

[0092] Next, consider the objective function (Equation 7) for each dimension d=1, 2, ..., D. d and M d are the input and output magnitudes, and p d is N d is a positive divisor of r d is the number of approximation terms that varies with the given tolerance ε. d and r d Since p takes discrete integer values, continuous optimization methods cannot be used directly. d N d The constraint that d Depending on the value of p dTo solve this problem, we can relax the constraints and d and r d Extend the domain of to the positive real numbers, and p d N d This removes the necessity to divide by . This leads to the following optimization problem. For brevity, we omit the subscript d below.

[0093] JPEG2026009798000028.jpg8150

[0094]

number

[0095] Because of the function ξ(ε,r), this optimization problem cannot be expressed in explicit form. Therefore, the control unit 330 approximates the constraint functions to reformulate this optimization problem into an unconstrained convex optimization problem.

[0096] If the tolerance is 0<ε<1, then mathematical formula 8 can be defined.

[0097]

number

[0098] JPEG2026009798000031.jpg17150

[0099] For a non-negative integer n, the nth power of x is expressed as:

[0100]

number

[0101] where T n (X) is a Chebyshev polynomial of degree n (if n is even, divide the coefficients of T0(x) by 2). Then,

[0102]

number

[0103] T for n-2k ≥ r n-2k Eliminating terms yields a Chebyshev approximation of order smaller than r. If η(r) is the maximum error of the approximation, then ξ(η(r),r)=c. Explicitly,

[0104]

number

[0105] Substituting n←n+2k, we can further express it as follows:

[0106]

number

[0107] JPEG2026009798000036.jpg26150

[0108] JPEG2026009798000037.jpg12150

[0109] An integer v is v≧w-1, and J v+1 (w) <j v (w) is maintained because the Bessel functions satisfy the recurrence relation shown in Equation 9.

[0110]

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[0111] JPEG2026009798000039.jpg15150

[0112]

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[0113] JPEG2026009798000041.jpg19150

[0114]

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[0115] Assume that η(r) is an upper bound because, for r ≥ 2, the approximate error function η(r) satisfies

[0116]

number

[0117] The upper bound can then be used to derive the relationship between the hyperparameters p and r.

[0118] Suppose that integer r0 ≥ 2 satisfies the equation U(r0) = ε. Then,

[0119]

number

[0120] where ξ(ε,r * )=c=ξ(η(r * ),r * ), the last equality holds. η(r) is non-decreasing by definition, so r * ≦r0. This means finding the solution to mathematical equation 11.

[0121]

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[0122] JPEG2026009798000046.jpg26150

[0123] JPEG2026009798000047.jpg13150

[0124]

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[0125] JPEG2026009798000049.jpg40150

[0126]

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[0127] According to the Banach Fixed-Point Theorem, the Unique Fixed Point converges to C. Set C0 = 0 and estimate C by the result of the second iteration of the algorithm.

[0128]

number

[0129] In mathematical formula 8, r * Assume that c~M / p by the definition of p. This leads to an approximate relationship between the hyperparameters p and r.

[0130]

number

[0131] In relation to the hyperparameter r, the hyperparameter p can be expressed as in Equation 12.

[0132]

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[0133] Using such a relationship, the objective function can be reduced to a functional form that depends only on r, thereby removing the inequality constraints.

[0134] JPEG2026009798000054.jpg8150

[0135]

number

[0136] JPEG2026009798000056.jpg12150

[0137] The convexity of the objective function ensures the convergence of quadratic optimization techniques such as the Newton algorithm. The optimal solution r that minimizes the objective function * After searching, we use the function p(r) to find the optimal p * =p(r * ) and p * Select the divisor of N that is closest to . This can overcome the problem of manual hyperparameter selection.

[0138] FIG. 4 is a flowchart illustrating the overall operation of the fast multidimensional partial Fourier transform device according to one embodiment, and FIG. 5 is a diagram illustrating tensors transformed by the fast multidimensional partial Fourier transform device according to one embodiment.

[0139] The overall operation of the fast multidimensional partial Fourier transform device is roughly divided into a configuration phase and a computation phase.

[0140] The configuration phase is a step of receiving input and output information and then setting up the hyperparameter optimization process and some of the necessary calculations in advance, and includes steps S410 to S415.

[0141] Steps S410 to S415 are steps of setting a plurality of hyperparameters to be used in the partial Fourier transform based on a tolerance for polynomial approximation and constraints on the degree of the polynomial. Setting the plurality of hyperparameters may include approximating and reconstructing constraints (constraint functions), and setting the plurality of hyperparameters from the reconstructed constraints by unconstrained convex optimization. Here, the plurality of hyperparameters may include a multidimensional degree, a multidimensional divisor, a multidimensional quotient, a multidimensional range tensor, optimal parentheses, or a combination thereof.

[0142] Step S410 receives information about the size N of the input array, the output domain (M, μ), and the tolerance ε. Step S411 sets a multidimensional order (r) using unconstrained convex optimization based on the constraints. Step S412 sets a multidimensional divisor (p) based on the size and multidimensional order of the array storing the multidimensional Fourier coefficients. Step S413 sets a multidimensional quotient (q) based on the size and multidimensional divisor of the array storing the multidimensional Fourier coefficients. Step S414 sets a multidimensional range tensor (B) based on the size, multidimensional order, and multidimensional quotient of the array storing the multidimensional Fourier coefficients. Step S415 sets an optimal bracket for an operation expressing the multidimensional Fourier coefficients using the multidimensional range tensor.

[0143] Algorithm 1 for the construction phase can be expressed in code as shown in Table 1.

[0144] [Table 1]

[0145] The inputs of Algorithm 1 are the input size N, the output domain M and μ, and the tolerance ε. The output is a tensor B for dimension d. (d) , divisor p d , quotient q d , order r d , and optimal parentheses.

[0146] The calculation phase is a step of outputting a Fourier coefficient array for a multidimensional domain through block decomposition, sequential tensor product, permutation, fast Fourier transform, and dot product calculation processes, and includes steps S420 to S426.

[0147] Steps S420 to S426 are steps of approximating and calculating multidimensional Fourier coefficients of a partial Fourier transform of multidimensional data based on a plurality of hyperparameters. The step of approximating and calculating multidimensional Fourier coefficients may include a step of performing a tensor transform based on a multivariate polynomial approximation on the multidimensional data using a multidimensional order, a multidimensional divisor, a multidimensional quotient, a multidimensional range tensor, an optimal bracket, or a combination thereof, and outputting approximated multidimensional Fourier coefficients.

[0148] Once the configuration phase is complete or the pre-set hyperparameters are confirmed, step S420 receives an arbitrary array 500 of size N and an output domain (M, μ). Step S421 performs block decomposition on the array 500 storing multidimensional Fourier coefficients based on a multidimensional divisor and a multidimensional quotient to generate a first tensor 510. Step S422 converts the first tensor 510 and a multidimensional range tensor into a second tensor 520 by performing a sequential tensor product based on optimal parentheses. Step S423 permutes the second tensor 520 based on the multidimensional order and converts it into a third tensor 530. Step S424 applies a fast Fourier transform to the third tensor 530 to convert it into a fourth tensor 540. In step S425, the fourth tensor 540, to which the fast Fourier transform has been applied, is dot-product-operated based on the multidimensional divisor and the multidimensional output domain to generate approximated multidimensional Fourier coefficients. In step S426, an array 550 storing the approximated multidimensional Fourier coefficients for the output domain is output.

[0149] Algorithm 2 for the computation phase can be expressed in code as shown in Table 2.

[0150] [Table 2]

[0151] JPEG2026009798000059.jpg13150

[0152] The algorithm for the fast multidimensional partial Fourier transform according to this embodiment can include Algorithm 1 and Algorithm 2, and can be called Auto-MPFT (Automatic Multidimensional Partial Fourier Transform).

[0153] The algorithm for fast multidimensional partial Fourier transform (Auto-MPFT) has three major advantages.

[0154] First, Auto-MPFT minimizes the time complexity of the fast multidimensional partial Fourier transform using optimal hyperparameters.

[0155] JPEG2026009798000060.jpg21150

[0156] Third, Auto-MPFT provides theoretical bounds on polynomial approximation. Given a sufficiently small error tolerance ε>0, the estimated Fourier coefficients of Equation 4 satisfy the following equation:

[0157]

number

[0158] By appropriately adjusting the tolerances, Auto-MPFT can be used to calculate Fourier coefficients to any numerical precision.

[0159] 6 to 8 are flowcharts of a fast multidimensional partial Fourier transform method according to one embodiment.

[0160] The fast multidimensional partial Fourier transform method according to the embodiment shown in Figures 6 to 8 includes steps that are processed in time series by the fast multidimensional partial Fourier transform device shown in Figures 1 to 5. Therefore, even though some content will be omitted below, the content described above regarding the fast multidimensional partial Fourier transform device shown in Figures 1 to 5 can also be applied to the fast multidimensional partial Fourier transform method according to the embodiment shown in Figures 6 to 8.

[0161] Referring to FIG. 6, in step S610, the fast multidimensional partial Fourier transform device sets a plurality of hyperparameters used for the partial Fourier transform based on the tolerance for polynomial approximation and constraints on the degree of the polynomial.

[0162] In step S620, the fast multidimensional partial Fourier transform device approximates and calculates multidimensional Fourier coefficients of the partial Fourier transform for the multidimensional data based on a plurality of hyperparameters.

[0163] The multiple hyperparameters may include a Multidimensional Degree, a Multidimensional Divisor, a Multidimensional Quotient, a Multidimensional Range Tensor, an Optimal Parenthesization, or a combination thereof.

[0164] The step of setting a plurality of hyperparameters (S610) may include a step of approximating and reconstructing the constraints, and setting a plurality of hyperparameters from the reconstructed constraints by unconstrained convex optimization.

[0165] Setting (S610) the plurality of hyperparameters may include setting a multidimensional order, a multidimensional divisor, a multidimensional quotient, a multidimensional range tensor, an optimal bracket, or a combination thereof based on the constraints.

[0166] The step of approximating and calculating multidimensional Fourier coefficients (S620) may include a step of performing a tensor transform based on a multivariate polynomial approximation on the multidimensional data using a multidimensional order, a multidimensional divisor, a multidimensional quotient, a multidimensional range tensor, an optimal bracket, or a combination thereof, and outputting approximated multidimensional Fourier coefficients.

[0167] Referring to FIG. 7, the step of setting a plurality of hyperparameters (S610) may include a step of setting a multidimensional order using unconstrained convex optimization based on constraints (S710); a step of setting a multidimensional divisor based on the size of an array storing multidimensional Fourier coefficients and the multidimensional order (S720); a step of setting a multidimensional quotient based on the size of an array storing multidimensional Fourier coefficients and the multidimensional divisor (S730); a step of setting a multidimensional range tensor based on the size of the array storing multidimensional Fourier coefficients, the multidimensional order, and the multidimensional quotient (S740); and a step of setting an optimal bracket for an operation expressing multidimensional Fourier coefficients using the multidimensional range tensor (S750).

[0168] Referring to FIG. 8, the step of approximating and calculating multidimensional Fourier coefficients (S620) may include the steps of: generating a first tensor by block decomposing an array storing multidimensional Fourier coefficients based on a multidimensional divisor and a multidimensional quotient (S810); converting the first tensor and a multidimensional range tensor into a second tensor by performing a sequential tensor product based on the optimal parentheses (S820); permuting the second tensor based on the multidimensional order and converting it into a third tensor (S830); applying a fast Fourier transform to the third tensor to convert it into a fourth tensor (S840); and outputting an array storing approximated multidimensional Fourier coefficients by performing a dot product of the fourth tensor to which the fast Fourier transform has been applied based on the multidimensional divisor and the multidimensional output domain (S850).

[0169] According to this embodiment, partial Fourier coefficients can be automatically and efficiently calculated using multidimensional data.

[0170] According to this embodiment, the computational cost can be significantly reduced by efficiently approximating trigonometric factors of multidimensional data using multivariate polynomial approximation, which maximizes performance by effectively utilizing tensor multiplication and multidimensional fast Fourier transform when processing multidimensional data.

[0171] According to this embodiment, an unconstrained convex optimization algorithm that automatically selects hyperparameters can be introduced to maintain optimal performance without manual adjustment by the user. This optimization algorithm enables explicit reconstruction of complex constraint functions using the Chebyshev approximation method and ensures efficient convergence using the Newton method.

[0172] This embodiment provides speeds up to 7.6 times faster than existing partial Fourier transforms while maintaining accuracy and significantly reducing the additional cost of hyperparameter search.

[0173] This embodiment can be used in the field of digital signal processing, which requires frequency domain processing of multidimensional data. Since this embodiment efficiently processes multidimensional data and automatically adjusts hyperparameters to significantly improve data processing efficiency, it can be applied to the artificial intelligence and machine learning industries, which require real-time processing of large amounts of data. It can be used in many cutting-edge technology fields, such as data center operation, cloud computing, real-time streaming services, geographic information systems, and autonomous vehicles. As methodologies using fast Fourier transforms for rapid learning, inference, and calculations increase in the field of machine learning, this embodiment can be used in spectral analysis techniques to increase the efficiency of convolutional neural networks (CNNs), which are widely used in visual video analysis, thereby reducing the total amount of computation.

[0174] The term "module" used in the above embodiments refers to software or hardware components such as FPGAs (field programmable gate arrays) or ASICs, and the "module" performs a certain function. However, the term "module" is not limited to software or hardware. A "module" may be configured to reside on an addressable storage medium or to execute one or more processors. Thus, by way of example, "module" includes components such as software components, object-oriented software components, class components, and task components, as well as processes, functions, attributes, procedures, subroutines, segments of program code, drivers, firmware, microcode, circuits, data, databases, data structures, tables, arrays, and variables.

[0175] The functionality provided within components and units may be combined into fewer components and units or separated into additional components and units.

[0176] Furthermore, the components and "units" may be implemented to implement one or more CPUs within a device or a secure multimedia card.

[0177] Meanwhile, the fast multidimensional partial Fourier transform method according to an embodiment of the present disclosure may also be embodied in the form of a computer-readable medium storing computer-executable instructions and data. Here, the instructions and data may be stored in the form of program code, which, when executed by a processor, generates a predetermined program module and performs a predetermined operation. The computer-readable medium may be any available medium accessible by a computer, including both volatile and nonvolatile media, and both separable and non-separable media. The computer-readable medium may also be a computer recording medium. The computer recording medium may include both volatile and non-volatile, separable and non-separable media embodied in any method or technology for storing information such as computer-readable instructions, data structures, program modules, or other data. For example, the computer recording medium may be a magnetic storage medium such as a hard disk drive (HDD) or solid-state drive (SSD), an optical storage medium such as a CD, DVD, or Blu-ray disc, or a memory included in a server accessible via a network.

[0178] Furthermore, the fast multidimensional partial Fourier transform method according to an embodiment described herein may be embodied as a computer program (or a computer program product) including computer-executable instructions. The computer program includes programmable machine instructions to be processed by a processor, and may be embodied in a high-level programming language, an object-oriented programming language, an assembly language, a machine language, or the like. The computer program may be recorded on any type of computer-readable recording medium (e.g., memory, a hard disk, a magnetic / optical medium, or a solid-state drive (SSD)).

[0179] Therefore, the fast multidimensional partial Fourier transform method according to one embodiment described herein can be implemented by executing the above-described computer program on a computing device. The computing device can include at least a processor, a memory, a storage device, a high-speed interface connecting to the memory and a high-speed expansion port, and a low-speed interface connecting to a low-speed bus and the storage device. Each of these components is connected to each other using various buses and can be mounted on a common motherboard or in other suitable manners.

[0180] Here, the processor may process instructions within a computing device. Such instructions may include instructions stored in a memory or storage device for displaying graphic information to provide a GUI (Graphical User Interface) on an external input and output device, such as a display connected to a high-speed interface. In other embodiments, multiple processors and / or multiple buses may be used, along with multiple memories and memory types, as appropriate. Also, the processor may be implemented as a chipset consisting of chips containing multiple independent analog and / or digital processors.

[0181] Also, memory stores information within a computing device. As an example, memory may be comprised of a volatile memory unit or collection thereof. As another example, memory may be comprised of a non-volatile memory unit or collection thereof. Memory may also be in other forms of computer-readable media, such as magnetic or optical disks.

[0182] The storage device can provide a large amount of storage space to a computing device. The storage device may be a computer-readable medium or a configuration that includes such a medium, such as a device in a Storage Area Network (SAN) or other configuration, and may be a floppy disk drive, hard disk drive, optical disk drive, tape drive, flash memory, or other similar semiconductor memory device or device array.

[0183] The above-described embodiments are merely illustrative, and those skilled in the art will understand that the above-described embodiments may be easily modified into other specific forms without changing the technical ideas or essential features of the above-described embodiments. Therefore, it should be understood that the above-described embodiments are illustrative in all respects and are not limiting. For example, each component described as a single component may be implemented in a distributed form, and similarly, each component described as a distributed component may be implemented in a combined form.

[0184] The scope of protection sought by this specification is determined by the claims set forth below rather than the above detailed description, and all modifications or variations derived from the meaning and scope of the claims and their equivalent concepts should be construed as being included within the scope of the present invention. [Explanation of symbols]

[0185] 300 High-speed multidimensional partial Fourier transform device 310 Input / output section 320 memory 330 Control Unit 340 Communications Department

Claims

1. A fast multidimensional partial Fourier transform method executed by a fast multidimensional partial Fourier transform device, comprising: setting a plurality of hyperparameters used in the partial Fourier transform based on a tolerance for the polynomial approximation and constraints on the degree of the polynomial; and calculating, by approximating, multidimensional Fourier coefficients of the partial Fourier transform for the multidimensional data based on the plurality of hyperparameters.

2. The step of setting the plurality of hyperparameters includes:

2. The fast multidimensional partial Fourier transform method according to claim 1, further comprising a step of approximating and reconstructing the constraints, and setting the plurality of hyperparameters from the reconstructed constraints by unconstrained convex optimization.

3. 2. The fast multidimensional partial Fourier transform method of claim 1, wherein the plurality of hyperparameters comprises a multidimensional degree, a multidimensional divisor, a multidimensional quotient, a multidimensional range tensor, an optimal parentheses, or a combination thereof.

4. The step of setting the plurality of hyperparameters includes:

4. The fast multidimensional partial Fourier transform method of claim 3, further comprising setting the multidimensional order, the multidimensional divisor, the multidimensional quotient, the multidimensional range tensor, the optimal bracket, or a combination thereof based on the constraints.

5. The step of setting the plurality of hyperparameters includes: setting the multidimensional order using unconstrained convex optimization based on the constraints; setting the multidimensional divisor based on the size of an array storing the multidimensional Fourier coefficients and the multidimensional order; setting the multidimensional quotient based on the size of an array storing the multidimensional Fourier coefficients and a multidimensional divisor; setting the multidimensional range tensor based on the size of an array storing the multidimensional Fourier coefficients, the multidimensional order, and the multidimensional quotient; and setting the optimal bracket for an operation using the multidimensional range tensor to represent the multidimensional Fourier coefficients.

6. The step of calculating the multidimensional Fourier coefficients by approximation includes:

4. The fast multidimensional partial Fourier transform method of claim 3, further comprising: performing a tensor transform based on a multivariate polynomial approximation on the multidimensional data using the multidimensional order, the multidimensional divisor, the multidimensional quotient, the multidimensional range tensor, the optimal bracket, or a combination thereof, to output approximated multidimensional Fourier coefficients.

7. The step of calculating the multidimensional Fourier coefficients by approximation includes: generating a first tensor by block decomposing an array storing the multidimensional Fourier coefficients based on the multidimensional divisor and the multidimensional quotient; converting the first tensor and the multidimensional range tensor into a second tensor by performing a sequential tensor product based on the optimal bracket; Permuting the second tensor based on the multidimensional order to convert it into a third tensor; applying a fast Fourier transform to the third tensor to convert it into a fourth tensor; and performing a dot product operation on the fourth tensor to which the fast Fourier transform is applied based on the multidimensional divisor and the multidimensional output domain, and outputting an array storing approximated multidimensional Fourier coefficients.

8. A high-speed multidimensional partial Fourier transform device comprising: a control unit that sets a plurality of hyperparameters to be used in a partial Fourier transform based on a tolerance for polynomial approximation and constraints on the degree of the polynomial; and that calculates and approximates multidimensional Fourier coefficients of the partial Fourier transform for multidimensional data based on the plurality of hyperparameters.

9. A computer-readable recording medium having a program recorded thereon for executing the method of claim 1.

10. A computer program stored on a recording medium for executing the method of claim 1 when executed by a fast multidimensional partial Fourier transform device.

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