Fast Multidimensional Partial Fourier Transform Method and Apparatus for Assisting Automatic Hyperparameter Selection

Abstract

Provided are a high-speed multi-dimensional partial Fourier transform method and apparatus that automatically select hyperparameters used for polynomial approximation of the Fourier transform, and that utilize the polynomial approximation to quickly calculate some of the Fourier coefficients for multi-dimensional data while maintaining accuracy. 【Solution means】The high-speed multi-dimensional partial Fourier transform method executed by the high-speed multi-dimensional partial Fourier transform apparatus includes: setting a plurality of hyperparameters used for partial Fourier transform based on an allowable error for polynomial approximation and constraints on the degree of the polynomial; and approximating and calculating multi-dimensional Fourier coefficients of partial Fourier transform for multi-dimensional data based on the plurality of hyperparameters.

Description

Technical Field

[0001] The embodiments disclosed in this specification relate to a high-speed multidimensional partial Fourier transform method and apparatus, and more particularly, to a high-speed multidimensional partial Fourier transform method and apparatus that utilize polynomial approximation to quickly calculate a part of Fourier coefficients for multidimensional data while maintaining accuracy.

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

[0003] This research was conducted as a research result of the "Development of Flexible and Efficient Model Compression Technology (SW Star Lab) 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 & communications Technology Planning and Evaluation (IITP).

[0004] This research was conducted as a research result of the "Support for AI Graduate School (Seoul National University)" project (IITP-2021-0-01343) and the "Research and Development of AI Innovation Hub" project (IITP-2021-0-02068) of the Information and Communication Broadcasting Human Resources Development project of the Ministry of Science and ICT and the Institute for Information & communications Technology Planning and Evaluation (IITP).

Background Art

[0005] The Fourier transform (FT) is a transform that decomposes an input signal into various frequency components and is represented 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 operations 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 mainly used to calculate discrete Fourier coefficients is the Fast Fourier Transform (FFT). The fast Fourier transform uses an inefficient method of calculating all the Fourier coefficients and then selecting only some of the necessary coefficients and discarding the rest.

[0008] The Partial Fourier Transform (PFT), introduced to solve such inefficient problems, is an algorithm that calculates only some of the discrete Fourier coefficients of the data and efficiently analyzes the characteristics of the data.

[0009] Existing partial Fourier transforms have the limitation that they are only useful when the number of Fourier coefficients that need to be calculated is very small compared to the size of the input data. In particular, existing partial Fourier transforms do not consider the input of multi-dimensional data and do not consider how to optimize the calculation process for some of the Fourier coefficients for multi-dimensional data.

[0010] For reference, Patent Document 1 is an invention related to an efficient implementation method of multi-dimensional 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 about the calculation of the fast Fourier transform and do not provide a fast partial Fourier transform technology for multi-dimensional data.

Prior Art Documents

Patent Documents

[0011]

Patent Document 1

Patent Document 2

Patent Document 3

Non-Patent Documents

[0012]

Non-Patent Document 1

Summary of the Invention

Problems to be Solved by the Invention

[0013] The embodiments disclosed herein automatically select hyperparameters used for polynomial approximation of Fourier transform, and provide a fast multi-dimensional partial Fourier transform method and apparatus that utilize polynomial approximation to quickly calculate a part of Fourier coefficients for multi-dimensional data while maintaining accuracy.

[0014] Other objects and advantages of the present invention will be understandable from the following description and will be more clearly understood by one example. Also, it will be easily understood that the objects and advantages of the present invention can be realized by the means and combinations thereof shown in the claims.

Means for Solving the Problems

[0015] As a technical means for achieving the above-described technical problem, a high-speed multi-dimensional partial Fourier transform method executed by a high-speed multi-dimensional partial Fourier transform device includes a step of setting a plurality of hyperparameters used for partial Fourier transform based on an allowable error for polynomial approximation and a constraint on the degree of a polynomial, and a step of approximating and calculating multi-dimensional Fourier coefficients of the partial Fourier transform for multi-dimensional data based on the plurality of hyperparameters.

[0016] According to another embodiment, a high-speed multi-dimensional partial Fourier transform device includes a control unit that sets a plurality of hyperparameters used for partial Fourier transform based on an allowable error for polynomial approximation and a constraint on the degree of a polynomial, and approximates and calculates multi-dimensional Fourier coefficients of the partial Fourier transform for multi-dimensional data based on the plurality of hyperparameters.

[0017] According to still another embodiment, a recording medium is a computer-readable recording medium on which a program for executing a high-speed multi-dimensional partial Fourier transform method is recorded.

[0018] According to still another embodiment, a computer program is a computer program that is executed by a high-speed multi-dimensional partial Fourier transform device and is stored in a recording medium to execute a high-speed multi-dimensional partial Fourier transform method.

Advantages of the Invention

[0019] According to any one of the above-described problem-solving means, a high-speed multi-dimensional partial Fourier transform method and device can be presented that approximate trigonometric function factors of multi-dimensional data by multivariate polynomial approximation and efficiently calculate a part of Fourier coefficients of multi-dimensional data by utilizing tensor transformation and tensor product when processing multi-dimensional data.

[0020] Moreover, according to any one of the above-described problem-solving means, explicit reconstruction of complex constraint conditions can be achieved by polynomial approximation, and using unconstrained convex optimization, even when the sizes of the input and output for multi-dimensional data processing are changed, a high-speed multi-dimensional fractional Fourier transform method and apparatus capable of automatically selecting optimal hyperparameters can be presented. The effects obtained in the disclosed embodiments are not limited to the effects mentioned above, and other effects not mentioned will be clearly understandable to those skilled in the technical field to which the disclosed embodiments belong from the following description.

Brief Description of the Drawings

[0021] Hereinafter, the attached drawings illustrate preferred embodiments disclosed in this specification and serve to better understand the technical idea disclosed in this specification together with the specific content for implementing the invention. Therefore, the content disclosed in this specification should not be construed as being limited only to the matters described in those drawings.

[0022]

Figure 1

Figure 2

Figure 3

Figure 4

Figure 5

Figure 6

Figure 7

Figure 8

Mode for Carrying Out the Invention

[0023] Hereinafter, various embodiments will be described in detail based on the accompanying drawings. The embodiments described below can also be implemented in various forms. In order to more clearly explain the features of the embodiments, detailed descriptions of matters well known to those having ordinary knowledge in the technical field to which the following embodiments belong are omitted. And, parts not related to the description of the embodiments in the drawings are omitted, and similar reference numerals are given to similar parts throughout the specification.

[0024] Throughout the specification, when it is said that a certain configuration is "connected" to another configuration, this includes not only the case where it is "directly connected", but also the case where it is "connected with other configurations interposed therebetween". Also, when it is said that a certain configuration "includes" another configuration, this means that, unless otherwise specified to the contrary, it does not exclude other configurations, and can further include other configurations.

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

[0026] "Hyperparameter" is a parameter for setting details necessary for calculating a polynomial approximation for approximating the multi-dimensional Fourier coefficients of the partial Fourier transform for multi-dimensional data.

[0027] "Tensor" is a data form that represents a multi-dimensional array. Tensor data can be used to represent and perform operations on various data such as text, images, sounds, and videos on a computer. There are various ways to represent tensors on a computer, typically including the rank indicating the dimension, the shape indicating the number of elements corresponding to each dimension, and the data type regarding the values. For example, 0-dimensional data is a 0-dimensional tensor or a scalar, an array of 1-dimensional data is a 1-dimensional tensor or a vector, an array of 2-dimensional data is a 2-dimensional tensor or a matrix, and an array of data with 3 or more dimensions can be classified as a multi-dimensional tensor or an N-dimensional tensor. Multiple tensors can be operated on each other for data processing.

[0028] Hereinafter, examples will be described in detail based on the accompanying drawings.

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

[0030] The multiple Fourier maps 121, 122, 123 shown in Figure 1 are visualizations of the Fourier coefficients for the multiple input images 111, 112, 113, and are maps representing the Fourier coefficients in log amplitude. In the multiple Fourier maps 121, 122, 123, except for the low-frequency part in the center and periphery, the Fourier coefficients are mostly close to 0. This indicates that by concentrating only on the non-zero coefficient parts and omitting the calculation of unnecessary coefficients, computational efficiency can be obtained. Most actual data such as time-series data, images, and videos have a very compressed representation in the frequency domain. It is necessary to efficiently calculate only some of the Fourier coefficients by utilizing the energy compression characteristics of the data.

[0031] Figure 2 is a diagram illustrating the ways in which an existing partial Fourier transform and a high-speed multi-dimensional partial Fourier transform device according to an embodiment process 2D inputs respectively.

[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 large number of one - dimensional partial Fourier transforms to each dimension. So, a cost of S·(S + TlogT)+T·(S + TlogT)~S 2 + STlogT is required. The high - speed multi - dimensional partial Fourier transform device according to one embodiment applies the partial Fourier transform to the entire input once. So, a cost of S 2 + T 2 logT 2 ~S 2 + T 2 logT is required. Here, since T << S, the high - speed multi - dimensional partial Fourier transform device according to one 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, in order to use the existing partial Fourier transform algorithm according to Non-Patent Document 1, the user must directly select an appropriate divisor for N. Since the overall performance of the existing partial Fourier transform algorithm varies greatly depending on the divisor value, it is important to select the optimal divisor each time the size of the input or output is changed. However, since the existing partial Fourier transform algorithm does not provide an option to automatically search for the optimal value, in the worst case, the user has to go through trial and error for all divisors. In particular, when the input is multi-dimensional, the situation becomes even worse because the search space of the hyperparameters grows geometrically with the dimension. The high-speed multi-dimensional partial Fourier transform device according to an embodiment can greatly reduce the cost due to hyperparameter search by using an algorithm based on Convex Optimization to automatically search for the optimal value of the divisor.

[0036] To overcome such problems, the high-speed multi-dimensional partial Fourier transform device according to this embodiment automatically selects hyperparameters and efficiently and accurately calculates a part of the Fourier coefficients for multi-dimensional data based on the automatically selected hyperparameters. The high-speed multi-dimensional partial Fourier transform device according to this embodiment performs two major operations.

[0037] First, the high-speed multi-dimensional partial Fourier transform device according to this embodiment calculates the partial Fourier coefficients using a pre-calculation technique based on the multivariate polynomial approximation of trigonometric functions. The high-speed multi-dimensional partial Fourier transform device uses a multivariate polynomial to approximate the set of trigonometric factors of the partial Fourier transform. The high-speed multi-dimensional partial Fourier transform device decomposes the partial Fourier transform into small sub-blocks and approximates some trigonometric functions with Chebyshev Polynomials to reduce the computational cost.

[0038] The high-speed multi-dimensional partial Fourier transform device according to this embodiment efficiently calculates partial Fourier coefficients by utilizing an arrangement matrix multiplication optimized for multi-dimensional data types and a fast Fourier transform algorithm. The high-speed multi-dimensional partial Fourier transform device uses the Cooley-Tukey algorithm to find a set of Smooth Twiddle Factors in the multi-dimensional partial Fourier transform. Then, a multivariate polynomial is used to approximate the twiddle factors. This can significantly reduce the time cost by decomposing the calculation of the partial Fourier coefficients into matrix multiplication and the multi-dimensional fast Fourier transform of small lower-order blocks of the input.

[0039] The high-speed multi-dimensional partial Fourier transform device according to this embodiment is superior in performance to existing partial Fourier transform algorithms, does not reduce accuracy, and improves the processing speed by up to 7.6 times.

[0040] Second, the high-speed multi-dimensional partial Fourier transform device according to this embodiment automatically searches for optimal hyperparameters and calculates the degree of the approximation polynomial. By automatically selecting hyperparameters without manual hyperparameter search by an algorithm based on convex optimization, the additional cost required for hyperparameter search is significantly reduced.

[0041] The optimal hyperparameters derived by the high-speed multi-dimensional partial Fourier transform device according to this embodiment are values that minimize the time complexity of the multi-dimensional partial Fourier transform operation. Although the constraint function of the optimization problem cannot be expressed in an explicit form, the high-speed multi-dimensional partial Fourier transform device induces an explicit reconstruction of the constraint function based on Chebyshev polynomial approximation to approximate the complex constraint function of the multi-dimensional partial Fourier transform to an unconstrained convex optimization problem. Thereby, an optimal hyperparameter can be efficiently searched using numerical analysis such as Newton's Method.

[0042] FIG. 3 is a block diagram for explaining the functional configuration of a high-speed multi-dimensional partial Fourier transform device according to an embodiment.

[0043] Referring to FIG. 3, a high-speed multi-dimensional partial Fourier transform apparatus 300 according to an embodiment can include an input / output unit 310, a memory 320, a control unit 330, and a communication unit 340. Some configurations can be omitted as necessary.

[0044] The input / output unit 310 can include an input unit for receiving input from a user and an output unit for displaying information such as the execution result of an operation or the state of the high-speed multi-dimensional partial Fourier transform apparatus 300. That is, the input / output unit 310 is a configuration for receiving input of data and outputting the result of arithmetic processing thereof. The high-speed multi-dimensional partial Fourier transform apparatus 300 according to the embodiment can receive a high-speed multi-dimensional partial Fourier transform request or the like via the input / output unit 310.

[0045] The memory 320 is a configuration capable of storing files and programs and can be constituted by various types of memories. In particular, the memory 320 can store data and programs that enable the control unit 330 described later to execute operations for high-speed multi-dimensional partial Fourier transform according to the algorithms presented below.

[0046] The memory 320 can store a plurality of hyperparameters used for high-speed multi-dimensional partial Fourier transform. The memory 320 can store partial Fourier coefficients for multi-dimensional data calculated based on the plurality of hyperparameters.

[0047] The control unit 330 is a configuration including at least one processor such as a CPU or a GPU and can control the overall operation of the high-speed multi-dimensional partial Fourier transform apparatus 300. That is, the control unit 330 can control other configurations included in the high-speed multi-dimensional partial Fourier transform apparatus 300 to execute operations for high-speed multi-dimensional partial Fourier transform. The control unit 330 can execute operations for approximating multi-dimensional Fourier coefficients by high-speed multi-dimensional partial Fourier transform according to the algorithms presented below by executing the programs stored in the memory 320.

[0048] The communication unit 340 can perform wired and wireless communications with other devices or networks. For example, the communication unit 340 can receive a plurality of hyperparameters and transmit multidimensional Fourier coefficients.

[0049] For this purpose, the communication unit 340 can include a communication module that supports at least one of various wired and wireless communication methods, and the communication module can be embodied in the form of a chipset. The mobile communication or wireless communication supported by the communication unit 340 can be, for example, an N-generation mobile communication protocol, WiFi (Wireless Fidelity), Wi-Fi Direct, Bluetooth (registered trademark), 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 a plurality of hyperparameters used for partial Fourier transform based on the allowable error for polynomial approximation and the constraints on the degree of the polynomial.

[0051] The control unit 330 defines the optimal hyperparameters as values that minimize the time complexity of the algorithm. Although the constraint function of such an optimization problem cannot be expressed in an explicit form, the control unit 330 reconstructs the constraint function by Chebyshev approximation to derive an unconstrained convex optimization problem in order to solve this problem. Since such an approach induces the convexity of the objective function, the optimal hyperparameters can be efficiently searched by numerical interpretation such as Newton's Method. After searching for the optimal solution that minimizes the objective function, the control unit 330 can automatically derive the optimal hyperparameters by approximating the optimal hyperparameters using a specific function.

[0052] The control unit 330 efficiently calculates partial Fourier coefficients through multi-dimensional partial Fourier transform. The control unit 330 approximates and calculates the multi-dimensional Fourier coefficients of the partial Fourier transform for multi-dimensional 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 less volatility in the multi-dimensional partial Fourier transform. Subsequently, Chebyshev polynomials are used to approximate the trigonometric functions. Such a method decomposes the calculation of partial Fourier coefficients into matrix multiplication and multi-dimensional fast Fourier transform for small sub-blocks of the input, greatly reducing the time cost.

[0054] The control unit 330 divides into a Configuration Phase and a Computation Phase to calculate the multi-dimensional partial Fourier transform coefficients at high speed. The control unit 330 performs a multivariate Chebyshev polynomial approximation for the trigonometric functions of the partial Fourier transform in the configuration phase and executes a sequential tensor product with the input sub-blocks. Also, it automatically searches for the optimal hyperparameters and calculates the degree of the approximation polynomial. The control unit 330 utilizes an arrangement matrix multiplication optimized for the multi-dimensional data type and a fast Fourier transform algorithm in the Computation Phase.

[0055] JPEG0007705540000002.jpg26170

[0056]

Number

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

[0058] JPEG0007705540000004.jpg33150

[0059] The control unit 330 can decompose Mathematical Formula 1 into Mathematical Formula 2 by using the Cooley - Tukey algorithm that recursively divides N in half and performs divide - and - conquer until a signal of length 2 is obtained.

[0060]

Number

[0061] JPEG0007705540000006.jpg11150

[0062] JPEG0007705540000007.jpg19150

[0063]

Number

[0064] JPEG0007705540000009.jpg23169

[0065] JPEG0007705540000010.jpg41150

[0066]

Number

[0067] Such polynomials have uniqueness and existence.

[0068] The control unit 330 uses the Chebyshev approximation algorithm to calculate the optimal polynomial approximation. The Chebyshev polynomials are used because of their solid theoretical foundation, which is widely applicable to achieving the optimal approximation with respect to the uniform norm and includes contributing to the derivation of the error bound. Other types of orthogonal polynomials that can improve accuracy and efficiency when necessary can also be applied.

[0069] When the control unit 330 is given a tolerance ε > 0 and a positive integer r, it defines the range for the origin ξ(ε, r) such that the exponential function e πix can be approximated by a polynomial of degree less than r that has an approximation bound ε.

[0070]

Number

[0071] JPEG0007705540000013.jpg14150

[0072] If the control unit 330 is given a tolerance ε > 0, for each d, it can find a positive integer r d ) ≥ M d / P d that satisfies the condition. d

[0073] JPEG0007705540000014.jpg36150

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

[0075]

Number

[0076] JPEG0007705540000016.jpg30142

[0077] ​ JPEG0007705540000017.jpg15150

[0078]

Number

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

[0080] When (N, M, μ, ε) is given, the control unit 330 can pre-calculate the optimal parentheses in the configuration phase and bypass the parentheses problem in the calculation phase.

[0081] JPEG0007705540000019.jpg34150

[0082]

Number

[0083] JPEG0007705540000021.jpg18150

[0084]

Number

[0085] JPEG0007705540000023.jpg23150

[0086] The control unit 330 provides an algorithm based on convex optimization for selecting the optimal hyperparameters of the fast multi-dimensional partial Fourier transform. The control unit 330 approximates the constraint function to convert the optimization problem of the fast multi-dimensional partial Fourier transform (Problem 1) into an unconstrained convex optimization problem (Problem 2).

[0087] The optimal hyperparameters minimize the time complexity of the fast multi-dimensional partial Fourier transform. The control unit 330 sets a time cost function. Since the construction phase only includes data-independent processes, only the calculation phase for the time cost is considered. For simplicity, the following notations are used. For d = 1, 2, …, D, N = Π d N d 、M = Π d M d 、p = Π d p d 、q = Π d q d 、r = Π d r d are as follows.

[0088] JPEG0007705540000024.jpg18150

[0089]

Number

[0090] JPEG0007705540000026.jpg23150

[0091]

Number

[0092] Next, consider the objective function (Equation 7) for each dimension d = 1, 2, …, D. N d and M d are the sizes of the input and output, p d is a positive divisor of N d , r d is the number of approximation terms that varies according to a given tolerance ε. Since the variables p d and r d take discrete integer values, continuous optimization methods cannot be directly used. The constraint that p d divides N d depends on the value of N d and p dresults in an irregular domain. To solve such a problem, the constraints are relaxed to expand the domains of p d and r d to positive real numbers and remove the necessity for p d to divide N d . This leads to the following optimization problem. For the sake of brevity, the following subscript d notation is omitted.

[0093] JPEG0007705540000028.jpg8150

[0094]

Number

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

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

[0097]

Number

[0098] JPEG0007705540000031.jpg17150

[0099] For a non - negative integer n, the n - th power of x is expressed as follows.

[0100]

Number

[0101] Here, T n (X) is the Chebyshev polynomial of degree n (when n is even, the coefficient of T0(x) is divided by 2). Then, it is as follows.

[0102] [Mathematics]

[0103] T for n - 2k ≥ r n-2k If the term is deleted, a Chebyshev approximation of degree less than r is derived. Let η(r) be the maximum error of the approximation, then ξ(η(r), r) = c. Explicitly, it is as follows.

[0104] [Mathematics]

[0105] If we substitute n ← n + 2k, it can be further expressed as follows.

[0106] [Mathematics]

[0107] JPEG0007705540000036.jpg26150

[0108] JPEG0007705540000037.jpg12150

[0109] The integer v satisfies v ≥ w - 1, and J v+1 (w) < j v (w) is maintained. This is because the Bessel function satisfies a recurrence relation as shown in Equation (9).

[0110] [Mathematics]

[0111] JPEG0007705540000039.jpg15150

[0112] [Mathematics]

[0113] JPEG0007705540000041.jpg19150

[0114] [Number]

[0115] Assume that η(r) is the upper limit. This is because if r ≥ 2, the approximation error function η(r) satisfies the following equation.

[0116] [Number]

[0117] After that, the relationship between the hyperparameters p and r can be derived using the upper limit.

[0118] Assume that the integer r0 ≥ 2 satisfies the equation U(r0) = ε. Then, it is as follows.

[0119] [Number]

[0120] Here, since ξ(ε, r * ) = c = ξ(η(r * ), r * ), the last equation is maintained. Since η(r) does not decrease by definition, r * ≤ r0. This means finding the solution of Mathematical Equation 11.

[0121] [Number]

[0122] JPEG0007705540000046.jpg26150

[0123] JPEG0007705540000047.jpg13150

[0124]

Number

[0125] JPEG0007705540000049.jpg40150

[0126]

Number

[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 Equation 8, assume that c ~ M / p by the definition of r. 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]

Number

[0133] Using such a relationship, the objective function can be reduced to a form that depends only on r to remove the inequality constraint terms.

[0134] ​ JPEG0007705540000054.jpg8150

[0135]

Number

[0136] JPEG0007705540000056.jpg12150

[0137] The convexity of the objective function guarantees the convergence of second-order optimization techniques such as Newton's method. After searching for the optimal solution r that minimizes the objective function, the function p(r) is used to approximate the optimal p * = p(r * ) and select the divisor of N closest to p * . This can overcome the problem of manual selection of hyperparameters. *

[0138] FIG. 4 is a flowchart illustrating the overall operation of a high-speed multi-dimensional partial Fourier transform apparatus according to an embodiment, and FIG. 5 is a diagram illustrating a tensor transformed by the high-speed multi-dimensional partial Fourier transform apparatus according to an embodiment.

[0139] The overall operation of the high-speed multi-dimensional partial Fourier transform apparatus is roughly divided into a configuration phase and a computation phase.

[0140] The configuration phase is a stage where, after receiving input and output information, hyperparameter optimization progress and a part of the necessary calculations are preset, and includes steps S410 to S415.

[0141] The stages S410 to S415 are stages of setting a plurality of hyperparameters used for the partial Fourier transform based on the tolerance for polynomial approximation and the constraints on the degree of the polynomial. The stage of setting a plurality of hyperparameters can include a stage of approximating and reconstructing the constraints (constraint functions) and setting a plurality of hyperparameters from the reconstructed constraints by unconstrained convex optimization. Here, the plurality of hyperparameters can include a multidimensional degree, a multidimensional divisor, a multidimensional quotient, a multidimensional range tensor, an optimal parenthesization, or a combination thereof.

[0142] In stage S410, information inputs regarding the size N of the input array, the output region (M, μ), and the tolerance ε are received. In stage S411, the multidimensional degree (r) is set using unconstrained convex optimization based on the constraints. In stage S412, the multidimensional divisor (p) is set based on the size of the array for storing the multidimensional Fourier coefficients and the multidimensional degree. In stage S413, the multidimensional quotient (q) is set based on the size of the array for storing the multidimensional Fourier coefficients and the multidimensional divisor. In stage S414, the multidimensional range tensor (B) is set based on the size of the array for storing the multidimensional Fourier coefficients, the multidimensional degree, and the multidimensional quotient. In stage S415, the optimal parenthesization for the operation expressing the multidimensional Fourier coefficients using the multidimensional range tensor is set.

[0143] Algorithm 1 regarding 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 area M and μ, and the tolerance ε, and the output is the tensor B for the dimension d (d) , the divisor p d , the quotient q d , the degree r d , and the optimal parentheses.

[0146] The calculation phase is a phase that outputs the Fourier coefficient array for the multi-dimensional region in sequence by the block decomposition, sequential tensor product, order change, fast Fourier transform, and dot product operation processes, and includes the S420~S426 phases.

[0147] The S420~S426 phases are phases for approximating and calculating the multi-dimensional Fourier coefficients of the partial Fourier transform for multi-dimensional data based on a plurality of hyperparameters. The phase for approximating and calculating the multi-dimensional Fourier coefficients can include a phase of performing a tensor transformation based on a multivariate polynomial approximation on the multi-dimensional data using the multi-dimensional degree, multi-dimensional divisor, multi-dimensional quotient, multi-dimensional range tensor, optimal parentheses, or a combination thereof, and outputting the approximated multi-dimensional Fourier coefficients.

[0148] When the configuration phase is completed or the pre-set hyperparameters are confirmed, step S420 receives an arbitrary array 500 of size N and the input of the output region (M, μ). In step S421, the array 500 for storing the multidimensional Fourier coefficients is block decomposed based on the multidimensional divisor and the multidimensional quotient to generate the first tensor 510. In step S422, based on the optimal parentheses, the first tensor 510 and the multidimensional range tensor are sequentially operated by the sequential tensor product to be converted into the second tensor 520. In step S423, the second tensor 520 is reordered (Permute) based on the multidimensional degree to be converted into the third tensor 530. In step S424, the fast Fourier transform is applied to the third tensor 530 to be converted into the fourth tensor 540. In step S425, based on the multidimensional divisor and the multidimensional output region, the fourth tensor 540 to which the fast Fourier transform is applied is dot product operated to generate the approximated multidimensional Fourier coefficients. In step S426, the array 550 in which the approximated multidimensional Fourier coefficients for the output region are stored is output.

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

[0150]

Table 2

[0151] JPEG0007705540000059.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 (Auto-MPFT) for the fast multidimensional partial Fourier transform has three major advantages.

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

[0155] JPEG0007705540000060.jpg21150

[0156] Third, Auto-MPFT presents the theoretical bounds for polynomial approximation. Given a sufficiently small tolerance ε > 0, the estimated Fourier coefficients in Equation 4 satisfy the following equation.

[0157]

Equation

[0158] By appropriately adjusting the tolerance, Fourier coefficients can be calculated with arbitrary numerical precision using Auto-MPFT.

[0159] Figures 6 to 8 are flowcharts of a fast multi-dimensional partial Fourier transform method according to an embodiment.

[0160] The fast multi-dimensional 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 multi-dimensional partial Fourier transform apparatus shown in Figures 1 to 5. Therefore, even though the content to be omitted is mentioned below, the content described above about the fast multi-dimensional partial Fourier transform apparatus shown in Figures 1 to 5 can also be applied to the fast multi-dimensional partial Fourier transform method according to the embodiment shown in Figures 6 to 8.

[0161] Referring to Figure 6, in step S610, the fast multi-dimensional partial Fourier transform apparatus sets a plurality of hyperparameters used for partial Fourier transform based on the tolerance for polynomial approximation and the constraints on the degree of the polynomial.

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

[0163] The plurality of hyperparameters can include a multidimensional degree, a multidimensional divisor, a multidimensional quotient, a multidimensional range tensor, an optimal parenthesization, or a combination thereof.

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

[0165] The stage of setting a plurality of hyperparameters (S610) can include the stage of setting a multidimensional degree, a multidimensional divisor, a multidimensional quotient, a multidimensional range tensor, an optimal parenthesization, or a combination thereof based on the constraints.

[0166] The stage of approximating and calculating the multidimensional Fourier coefficients (S620) can include the stage of performing a tensor transformation based on a multivariate polynomial approximation on the multidimensional data using a multidimensional degree, a multidimensional divisor, a multidimensional quotient, a multidimensional range tensor, an optimal parenthesization, or a combination thereof, and outputting the 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 multi-dimensional degree using unconstrained convex optimization based on constraints (S710), a step of setting a multi-dimensional divisor based on the size of an array for storing multi-dimensional Fourier coefficients and the multi-dimensional degree (S720), a step of setting a multi-dimensional quotient based on the size of the array for storing multi-dimensional Fourier coefficients and the multi-dimensional divisor (S730), a step of setting a multi-dimensional range tensor based on the size of the array for storing multi-dimensional Fourier coefficients, the multi-dimensional degree, and the multi-dimensional quotient (S740), and a step of setting an optimal parenthesis for an operation expressing multi-dimensional Fourier coefficients using the multi-dimensional range tensor (S750).

[0168] Referring to FIG. 8, the step of approximately calculating multi-dimensional Fourier coefficients (S620) includes a step of block-decomposing (Block Decomposition) an array for storing multi-dimensional Fourier coefficients based on the multi-dimensional divisor and the multi-dimensional quotient to generate a first tensor (S810), a step of sequentially performing a tensor product (Sequential Tensor Product) operation on the first tensor and the multi-dimensional range tensor based on the optimal parenthesis to convert them into a second tensor (S820), a step of changing the order (Permute) of the second tensor based on the multi-dimensional degree to convert it into a third tensor (S830), a step of applying a fast Fourier transform to the third tensor to convert it into a fourth tensor (S840), and a step of performing a dot product operation on the fourth tensor to which the fast Fourier transform has been applied based on the multi-dimensional divisor and the multi-dimensional output region to output an array storing the approximated multi-dimensional Fourier coefficients (S850).

[0169] According to this embodiment, partial Fourier coefficients can be automatically and efficiently calculated with multi-dimensional data.

[0170] According to this embodiment, the computational cost can be greatly reduced by efficiently approximating the trigonometric function factor of multi-dimensional data by multivariate polynomial approximation. This maximizes the performance by effectively utilizing tensor multiplication and multi-dimensional fast Fourier transform when processing multi-dimensional data.

[0171] According to this embodiment, an unconstrained convex optimization algorithm for automatically selecting hyperparameters is introduced to maintain optimal performance without manual adjustment by the user. Such an optimization algorithm enables explicit reconstruction of complex constraint functions by the Chebyshev approximation method and ensures efficient convergence using Newton's method.

[0172] This embodiment provides a speed that is up to 7.6 times faster than the existing partial Fourier transform while maintaining accuracy, and greatly reduces the additional cost due to hyperparameter search.

[0173] This embodiment can be applied to the field of digital signal processing that requires frequency domain processing of multi-dimensional data. This embodiment can efficiently process multi-dimensional data and automatically adjust hyperparameters to greatly improve the efficiency of data processing, so it is applicable to the artificial intelligence and machine learning industries that must process large-scale data in real time. It can be applied in many cutting-edge technology fields such as data center operation, cloud computing, real-time streaming services, geographic information systems, and autonomous vehicles. As the methodology of using fast Fourier transform for fast learning, inference, and calculation increases 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, to reduce the total amount of calculations.

[0174] The term "~ section" used in the above embodiments means a software or a hardware component such as a field programmable gate array (FPGA) or an application specific integrated circuit (ASIC), and the "~ section" plays a certain role. However, the "~ section" is not meant to be limited to software or hardware. The "~ section" can also be configured to be in an addressable storage medium or to cause one or more processors to execute. Thus, as an example, the "~ section" includes components such as software components, object-oriented software components, class components, and task components, and processes, functions, attributes, procedures, subroutines, segments of program patent code, drivers, firmware, microcode, circuits, data, databases, data structures, tables, arrays, and variables.

[0175] The components and the functions provided within the "~ section" can be combined with a smaller number of components and "~ sections" or separated from additional components and "~ sections".

[0176] In addition, the components and the "~ section" can also be implemented to cause one or more CPUs within a device or a security multimedia card to execute.

[0177] On the one hand, the high-speed multi-dimensional fractional Fourier transform method according to an embodiment described herein can also be embodied in the form of a computer-readable medium that stores computer-executable instructions and data. Here, the instructions and data can be stored in the form of program code, and when executed by a processor, a predetermined program module can be generated to execute a predetermined operation. Further, the computer-readable medium may be any available medium accessible by a computer, including both volatile and non-volatile media, and both removable and non-removable media. Also, the computer-readable medium may be a computer recording medium. The computer recording medium can include any volatile and non-volatile, removable and non-removable media embodied by 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 can be a magnetic storage medium such as an HDD and an SSD, an optical recording medium such as a CD, a DVD, and a Blu-ray disc, or a memory included in a server accessible via a network.

[0178] Also, the high-speed multi-dimensional fractional Fourier transform method according to an embodiment described herein can also be embodied in a computer program (or computer program product) that includes computer-executable instructions. The computer program includes programmable machine instructions processed by a processor and can be embodied in a high-level programming language, an object-oriented programming language, an assembly language, or a machine language. Also, the computer program can be recorded on a computer-readable recording medium of a type (e.g., a memory, a hard disk, a magnetic / optical medium, or an SSD (Solid-State Drive)).

[0179] Therefore, the high-speed multi-dimensional fractional Fourier transform method according to an embodiment described in this specification can be implemented by a computer program as described above being executed by a computing device. The computing device can include at least a part of a processor, a memory, a storage device, a high-speed interface connected to the memory and a high-speed expansion port, and a low-speed interface connected to a low-speed bus and the storage device. Each of such components is connected to each other using various buses and can be mounted on a common motherboard or installed in other suitable ways.

[0180] Here, the processor can process instruction words within the computing device. Such instruction words can have, for example, instruction words stored in the memory or the storage device for displaying graphic information for providing a GUI (Graphic User Interface) on an external input and output device such as a display connected to the high-speed interface. As another example, multiple processors and / or multiple buses can be appropriately used together with multiple memories and memory forms. Also, the processor can be implemented as a chipset consisting of chips including multiple independent analog and / or digital processors.

[0181] In addition, the memory stores information within the computing device. As an example, the memory can be composed of volatile memory units or a set thereof. As another example, the memory can be composed of non-volatile memory units or a set thereof. Also, the memory can be other forms of computer-readable media such as, for example, magnetic or optical disks.

[0182] And the storage device can provide a large-capacity storage space for the computing device. The storage device may be a computer-readable medium or a configuration including such a medium, and can also include, for example, devices within a SAN (Storage Area Network) or other configurations, and may be a floppy disk device, a hard disk device, an optical disk device, or a tape device, a flash memory, or other semiconductor memory devices or device arrays similar thereto.

[0183] The above-described embodiments are for illustrative purposes, and those with ordinary knowledge in the technical field to which the above-described embodiments belong will be able to understand that it is possible to easily transform the above-described embodiments into other specific forms without changing the technical idea or essential features possessed by the above-described embodiments. Therefore, it must be understood that the foregoing embodiments are illustrative in all respects and not restrictive. For example, each component described as a single type can also be implemented in a distributed manner, and similarly, the components described as being distributed can also be implemented in a combined form.

[0184] The scope to be protected by this specification is determined by the claims described later rather than the detailed description above, and all changes or variations derived from the meaning and scope of the claims and their equivalent concepts shall be construed as being included in the scope of the present invention.

Description of Reference Numerals

[0185] 300 High-speed Multidimensional Partial Fourier Transform Device 310 Input / Output Unit 320 Memory 330 Control Unit 340 Communication Unit

Claims

**Claim 1** A fast multi-dimensional partial Fourier transform method executed by a fast multi-dimensional partial Fourier transform device, comprising: setting a plurality of hyperparameters used for partial Fourier transform based on an allowable error for polynomial approximation and constraints on the degree of the polynomial; approximating and calculating multi-dimensional Fourier coefficients of the partial Fourier transform for multi-dimensional data based on the plurality of hyperparameters. A fast multi-dimensional partial Fourier transform method. **Claim 2** The step of setting the plurality of hyperparameters includes: the fast multi-dimensional partial Fourier transform method according to claim 1, comprising the step of approximating and reconstructing the constraints, and setting the plurality of hyperparameters from the reconstructed constraints by unconstrained convex optimization (Unconstrained Convex Optimization). **Claim 3** The plurality of hyperparameters include a multi-dimensional degree (Multidimensional Degree), a multi-dimensional divisor (Multidimensional Divisor), a multi-dimensional quotient (Multidimensional Quotient), a multi-dimensional range tensor (Multidimensional Range Tensor), an optimal parenthesis (Optimal Parenthesization), or a combination thereof. The fast multi-dimensional partial Fourier transform method according to claim 1. **Claim 4** The step of setting the plurality of hyperparameters includes: the fast multi-dimensional partial Fourier transform method according to claim 3, comprising the step of setting the multi-dimensional degree, the multi-dimensional divisor, the multi-dimensional quotient, the multi-dimensional range tensor, the optimal parenthesis, or a combination thereof based on the constraints. **Claim 5** The step of setting the plurality of hyperparameters includes: setting the multi-dimensional degree using unconstrained convex optimization based on the constraints; setting the multi-dimensional divisor based on the size of an array for storing the multi-dimensional Fourier coefficients and the multi-dimensional degree; setting the multi-dimensional quotient based on the size of an array for storing the multi-dimensional Fourier coefficients and the multi-dimensional divisor; setting the multi-dimensional range tensor based on the size of an array for storing the multi-dimensional Fourier coefficients, the multi-dimensional degree, and the multi-dimensional quotient; setting the optimal parentheses for the operation expressing the multi-dimensional Fourier coefficients using the multi-dimensional range tensor; The high-speed multi-dimensional partial Fourier transform method according to claim 3, comprising:

6. The step of approximately calculating the multi-dimensional Fourier coefficients is executing a tensor transformation based on a multivariate polynomial approximation on the multi-dimensional data using the multi-dimensional degree, the multi-dimensional divisor, the multi-dimensional quotient, the multi-dimensional range tensor, the optimal parentheses, or a combination thereof, and outputting the approximated multi-dimensional Fourier coefficients; The high-speed multi-dimensional partial Fourier transform method according to claim 3, comprising:

7. The step of approximately calculating the multi-dimensional Fourier coefficients is generating a first tensor by block decomposing (Block Decomposition) an array for storing the multi-dimensional Fourier coefficients based on the multi-dimensional divisor and the multi-dimensional quotient; sequentially performing a sequential tensor product operation on the first tensor and the multi-dimensional range tensor based on the optimal parentheses to convert them into a second tensor; changing the order (Permute) of the second tensor based on the multi-dimensional degree to convert it into a third tensor; applying a fast Fourier transform to the third tensor to convert it into a fourth tensor; performing a dot product operation on the fourth tensor to which the fast Fourier transform has been applied based on the multi-dimensional divisor and the multi-dimensional output region, and outputting an array storing the approximated multi-dimensional Fourier coefficients; The high-speed multi-dimensional partial Fourier transform method according to claim 3, comprising:

8. a control unit that sets a plurality of hyperparameters used for the partial Fourier transform based on the tolerance for polynomial approximation and the constraints on the degree of the polynomial, and approximately calculates the multi-dimensional Fourier coefficients of the partial Fourier transform on the multi-dimensional data based on the plurality of hyperparameters; A high-speed multi-dimensional partial Fourier transform device.

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

10. A computer program stored in a recording medium for executing the method according to claim 1, which is executed by a high-speed multi-dimensional partial Fourier transform device.

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