Analysis device, method, and program

Abstract

An analysis device according to one embodiment of the present invention comprises: a cyclic symmetry dual problem solution unit for inputting, to a dual problem based on an optimal transport problem with strongly convex regularization term, a first and second vector that are a portion of the symmetric components of first and second input data that have cyclic symmetry as well as a cost matrix that is a portion of the symmetric components of third input data that has cyclic symmetry, and outputting an optimal solution to the dual problem; and an optimal transport problem solution unit for using the optimal solution outputted by the cyclic symmetry dual problem solution unit to find and output an optimal solution and optimal objective function value of the optimal transport problem with strongly convex regularization term.

Description

Analytical device, method and program

[0001] FIELD Embodiments of the present invention relate to an analytical device, a method, and a program.

[0002] The discrete optimal transportation problem is a problem of finding a transportation matrix that is a transportation route that minimizes the total cost when transporting the cost matrix from one probability vector to the other probability vector, when two probability vectors and a cost matrix between them are given (see, for example, non-patent documents 1 and 2).

[0003] Marco Cuturi. Sinkhorn distances: Lightspeed computation of optimal transport. In Advances in Neural Information Processing Systems, 2013. Jason Altschuler, Francis Bach, Alessandro Rudi, and Jonathan Niles-Weed, Massively scalable Sinkhorn distances via the Nystrom method, In Advances in Neural Information Processing Systems, 2019.

[0004] Here, we define the mathematical notation we will use. The set of non-negative real numbers is expressed as in (1) below, and the inner product is expressed as in (2) below.

[0005]

[0006] When a vector is expressed as in (3) below, the inner product of this vector is expressed as in (4) below.

[0007]

[0008] Furthermore, when a matrix is ​​expressed as in (5) below, the inner product of this matrix is ​​expressed as in (6) below.

[0009]

[0010] When a d-dimensional probability simplex is expressed as in (7) below, a d-dimensional vector in which all elements are 1 is expressed as in (8) below.

[0011]

[0012] The optimal transportation problem with a strongly convex regularization term is defined as follows: First, when two d-dimensional random vectors are given as shown in (9) below and a d×d-dimensional cost matrix is ​​given as shown in (10) below, the optimal transportation problem with a strongly convex regularization term is defined as a minimization problem as shown in (11) below.

[0013]

[0014] Here, for this optimal transportation problem, the transport matrix is ​​expressed as in (12) below, and φ is expressed as in (13) below. Hereinafter, the transport matrix will be referred to as "transport matrix T".

[0015]

[0016] When the value expressed by the following (14) is convex for μ>0, φ is called a strongly convex value.

[0017]

[0018] Furthermore, when x>0, the following (15) holds, and it is guaranteed that the transport matrix T is non-negative. φ(x)=+∞ (15) The optimal objective function value of the optimal transport problem with a strongly convex regularization term as shown in (11) above is called the optimal transport distance and is used in various application programs.

[0019] The optimal transportation problem with a strongly convex regularization term expressed by the above formula (11) can be solved faster than a problem without a strongly convex regularization term, as shown in, for example, the above-mentioned Non-Patent Document 1, but it is known that it still requires a long computation time. Furthermore, the above-mentioned Non-Patent Document 2 describes a method for quickly solving the optimal transportation problem with a strongly convex regularization term by assuming that the cost matrix, which is the input data for the optimal transportation problem with a strongly convex regularization term such as the above formula (11), can be approximated at a low rank and utilizing the results of this approximation. However, there is still no promising method for quickly solving the optimal transportation problem without approximation.

[0020] The present invention has been made in light of the above-mentioned circumstances, and its object is to provide an analytical device, method, and program that can solve optimal transportation problems quickly without approximation.

[0021] An analysis apparatus according to one embodiment of the present invention includes a cyclic symmetric dual problem solving unit that inputs first and second vectors, which are part of the symmetric components of first and second input data having cyclic symmetry, and a cost matrix, which is part of the symmetric components of third input data having cyclic symmetry, into a dual problem based on an optimal transportation problem with a strongly convex regularization term, and outputs an optimal solution to the dual problem; and an optimal transportation problem solving unit that uses the optimal solution output by the cyclic symmetric dual problem solving unit to find and output an optimal solution to the optimal transportation problem with a strongly convex regularization term and an optimal objective function value.

[0022] An analytical device according to one embodiment of the present invention includes: a cyclic symmetric dual problem solving unit that calculates first and second vectors from first and second input data having approximate cyclic symmetry, inputs the calculated first and second vectors and a cost matrix that is a portion of the symmetric components of third input data into a first dual problem based on an optimal transportation problem with a strongly convex regularization term, and outputs an optimal solution to the first dual problem; an original dual problem solving unit that converts the optimal solution output by the cyclic symmetric dual problem solving unit into an initial solution that assumes an ideal state of cyclic symmetry of the input data, and uses the initial solution to output an optimal solution to a second dual problem based on the optimal transportation problem with a strongly convex regularization term; and an optimal transportation problem solving unit that uses the optimal solution output by the original dual problem solving unit to find and output an optimal solution to the optimal transportation problem with a strongly convex regularization term and an optimal objective function value.

[0023] According to the present invention, the optimal transportation problem can be solved quickly without approximation.

[0024] Fig. 1 is a diagram showing an application example of an analysis device according to an embodiment of the present invention. Fig. 2 is a diagram showing an example of a procedure for a processing operation by an analysis device according to a first embodiment of the present invention. Fig. 3 is a diagram showing an application example of an analysis device according to a second embodiment of the present invention. Fig. 4 is a diagram showing an example of a procedure for a processing operation by an analysis device according to the second embodiment of the present invention. Fig. 5 is a block diagram showing an example of the hardware configuration of an analysis device according to an embodiment of the present invention.

[0025] An embodiment of the present invention will be described below with reference to the drawings. In this embodiment, a new assumption is made that the input data for the optimal transportation problem with a strongly convex regularization term as shown in (11) above has n-th order cyclic symmetry, and by utilizing this assumption, the optimal transportation problem can be solved quickly.

[0026] Specifically, the optimal transportation problem with a strongly convex regularization term as shown in (11) above is usually converted into a dual problem in which the number of variables is 2d and then solved. However, in this embodiment, the optimal transportation problem with a strongly convex regularization term as shown in (11) above is converted into a dual problem in which the number of variables is smaller than the usual 2d, as shown in (16) below, by taking into account the cyclic symmetry.

[0027]

[0028] Therefore, in this embodiment, by solving a dual problem in which such a small number of variables are handled instead of a normal dual problem, it becomes possible to obtain a solution at high speed.

[0029] As described above, in this embodiment, it is newly assumed that the input data a, b, and C of the optimal transportation problem with a strongly convex regularization term as shown in the above (11) have n-th order cyclic symmetry. Specifically, it is assumed that the input data a and b have a repetitive structure as shown in the following (17).

[0030]

[0031] Furthermore, the relationship expressed by the following (18) holds, and the value shown by the following (19) is an integer.

[0032]

[0033] In this embodiment, it is assumed that the cost matrix C has a block cyclic structure as shown in (20) below.

[0034]

[0035] Furthermore, the following relationship expressed by (21) holds true.

[0036]

[0037] First Embodiment Next, a first embodiment will be described. In this first embodiment, an algorithm will be described that can quickly solve an optimal transportation problem with a strongly convex regularization term, where the input data of the optimal transportation problem with a strongly convex regularization term as shown in (11) above satisfies the assumption that the input data has n-th order cyclic symmetry.

[0038] The optimization algorithm in the first embodiment is an optimization algorithm that can realize the following parts (a) and (b): (a) Two vectors α and β, which are part of the symmetric components of the input data a, b, and C, and a cost matrix C ０ , ..., C Ｎ-１ (b) an optimal transportation problem solving unit that uses the optimal solution output from the cyclic symmetric dual problem solving unit to find and output the optimal solution and optimal objective function value of the optimal transportation problem with a strongly convex regularization term as shown in (11) above.

[0039] Fig. 1 is a diagram showing an application example of an analysis device according to a first embodiment of the present invention. Fig. 2 is a diagram showing an example of a procedure of processing operations by the analysis device according to the first embodiment of the present invention. As shown in Fig. 1, the analysis device 100 according to the first embodiment of the present invention includes a cyclic symmetric dual problem solving unit 11 and an optimal transportation problem solving unit 12.

[0040] The individual components are described below. The cyclic symmetric dual problem solving unit 11 solves two vectors α and β, which are part of the input data a, b, and c, and a cost matrix C ０ , ..., C Ｎ-１ is used as input data for a dual problem expressed by the following (22) (S11), the dual problem is solved, and an optimal solution of this dual problem expressed by the following (23) is output (S12).

[0041]

[0042] Here, the value expressed by the following (24) is called the Fenchel conjugate, and the relationship expressed by the following (25) is defined.

[0043]

[0044] In this embodiment, the algorithm for solving the dual problem expressed by the above (22) is not particularly limited. As an example of this algorithm, for example, the alternating minimization algorithm mentioned in Chapter 17 of the document "Amir Beck and Luba Tetruashvili, On the convergence of block coordinate descent type methods, SIAM journal on Optimization, 23(4): 2037-2060, 2013." can be used. Specifically, the cyclic symmetric dual problem solving unit 11 fixes the value of z for the dual problem expressed by the above (22) and optimizes w. ｉ The partial differential with respect to is given by the following equation (26), and this w ｉ is optimal when the value obtained by the equation (26) is 0.

[0045]

[0046] Similarly, the cyclic symmetric dual problem solving unit 11 fixes the value of w for the dual problem expressed by the above (22) and optimizes z. By alternately repeating such various optimizations, the optimal solution w for the dual problem expressed by the above (22) is obtained. ＊ , z ＊ Output.

[0047] Next, the optimal transportation problem solving unit 12 first calculates the optimal solution w of the dual problem expressed by the above (22), which is the output from the cyclic symmetric dual problem solving unit 11. ＊ , z ＊ Using the above, the optimal transport matrix T ＊ is calculated from the following relational expression (27) (S13).

[0048]

[0049] Here, the relational expression shown in (27) above is defined as shown in (28) below, and the optimal transport matrix T＊ is expressed as in (30) below using values ​​as shown in (29) below.

[0050]

[0051] The optimal transportation problem solving unit 12 can calculate the optimal transportation distance, which is the optimal objective function value of the optimal transportation problem with a strongly convex regularization term as shown in the above (11), using the following equation (31). This value and the optimal transportation matrix T ＊ Output.

[0052]

[0053] Second Embodiment Next, a second embodiment will be described. In this second embodiment, an algorithm will be described that can quickly solve an optimal transportation problem with a strongly convex regularization term, in which the input data approximately satisfies the assumption that the input data of the optimal transportation problem with a strongly convex regularization term, as shown in (11) above, has n-th order cyclic symmetry.

[0054] The algorithm in the second embodiment is an algorithm that can realize the following parts (a) to (c): (a) Calculate two vectors α and β from input data a and b, and calculate a cost matrix C that is a part of the symmetric components of these vectors α and β and input data C. ０ , ..., C Ｎ-１ (b) an original dual problem solving unit that converts the optimal solution output from the cyclic symmetric dual problem solving unit into an initial solution, solves the dual problem converted from the optimal transportation problem with strongly convex regularization terms as shown in (11) above, and outputs an optimal solution to the dual problem; and (c) an optimal transportation problem solving unit that uses the optimal solution output from the original dual problem solving unit to find and output an optimal solution and an optimal objective function value for the optimal transportation problem with strongly convex regularization terms, where the input data has approximate cyclic symmetry.

[0055] Fig. 3 is a diagram showing an application example of an analysis apparatus according to a second embodiment of the present invention. Fig. 4 is a diagram showing an example of a procedure of processing operations by the analysis apparatus according to the second embodiment of the present invention. As shown in Fig. 3, the analysis apparatus 100a according to the second embodiment of the present invention includes a cyclic symmetric dual problem solving unit 11a, an optimal transportation problem solving unit 12a, and an original dual problem solving unit 13a.

[0056] Each component is described below. The cyclically symmetric dual problem solving unit 11a solves an optimal transportation problem for input data that assumes ideal cyclic symmetry. This aims to obtain an initial solution that is useful for solving an optimal transportation problem with a strongly convex regularization term, such as that shown in (11) above, in which the input data has approximate cyclic symmetry. This initial solution is not the optimal solution because it assumes an ideal state of cyclic symmetry, but it is a solution that is expected to be somewhat close to the optimal solution.

[0057] Specifically, the cyclic symmetric dual problem solving unit 11a calculates two vectors α and β from input data a and b, and calculates a cost matrix C, which is a part of the symmetric components of the input data C, from these vectors α and β. ０ , ..., C Ｎ-１ are used as input data for the dual problem (S21), the dual problem is solved, and an optimal solution of the dual problem, as expressed by the following (32), is output (S22).

[0058]

[0059] First, the cyclic symmetric dual problem solving unit 11a calculates two vectors α and β from input data a and b that approximately satisfy the assumption that the input data of the optimal transportation problem with a strongly convex regularization term as shown in the above (11) has n-th order cyclic symmetry, as shown in the following (33).

[0060]

[0061] Thereafter, the cyclic symmetric dual problem solving unit 11a performs the same processing as the cyclic symmetric dual problem solving unit 11 in the first embodiment using the above calculation results to obtain the optimal solution w ＊ , z ＊ Output.

[0062] Next, the original dual problem solving unit 13a newly considers a dual problem of the original strongly convex regularized optimization problem as shown in the above (11). This dual problem can be expressed as the following (34).

[0063]

[0064] As in the case of the cyclic symmetric dual problem solving unit 11 described in the first embodiment, if an alternating minimization algorithm or the like is used to solve the dual problem of (34) above, f of the dual problem of (34) above can be calculated as follows: ｉ The partial differential with respect to is given as follows (35), and this f ｉ is optimal when the value obtained by the formula (35) is 0.

[0065]

[0066] Similarly, the original dual problem solving unit 13a fixes the value of f for the dual problem expressed by (34) above and optimizes g. By alternately repeating such various optimizations, the original dual problem solving unit 13a finds the optimal solution f for the dual problem expressed by (34) above. ＊ , g ＊ Output.

[0067] However, when the alternating minimization algorithm for f and g is performed, the initial solutions for f and g are the optimal solutions w ＊ , z ＊ That is, the original dual problem solving unit 13a calculates the optimal solution w output from the cyclic symmetric dual problem solving unit 11a as follows: ＊ , z ＊ The initial solution is transformed from ＊ , g ＊ is output (S23).

[0068]

[0069] In this way, in the second embodiment, the alternating minimization algorithm is started from an initial solution that takes into account the ideal cyclic symmetry of the input data, rather than from an arbitrary initial solution. This makes it possible to efficiently find an optimal solution to the optimal transportation problem with a strongly convex regularization term, such as that shown in (11) above, where the input data has approximate cyclic symmetry.

[0070] Next, the optimal transportation problem solving unit 12a first calculates the optimal solution f of the dual problem expressed by the above (34), which is the output from the original dual problem solving unit 13a. ＊ , g ＊ Using the above, the optimal transport matrix T ＊ is calculated from the following relational expression (37).

[0071]

[0072] Here, the relational expression shown in the above (37) is defined as shown in the following (38).

[0073]

[0074] Furthermore, the optimal transportation problem solving unit 12a can calculate the optimal objective function value of the optimal transportation problem with a strongly convex regularization term as shown in the above (11) using the above equation as shown in the above equation (31) explained in the first embodiment, and calculate this value and the optimal transportation matrix T ＊ is output (S24).

[0075] Third Embodiment Next, a third embodiment will be described. In this third embodiment, the strongly convex function φ in the optimal transportation problem with strongly convex regularization terms as shown in (11) above in the first embodiment is given in the form of the following (39).

[0076]

[0077] However, in the above (39), λ>0. In this case, the strongly convex optimal transportation problem shown in the above (11) coincides with the entropy-regularized optimal transportation problem, which is an important subclass of the strongly convex optimal transportation problem disclosed in, for example, Non-Patent Document 1.

[0078] In this third embodiment, an algorithm is described that can quickly solve the entropy regularized optimal transportation problem, which is an optimal transportation problem when the input data for the strongly convex optimal transportation problem as shown in (11) above satisfies the assumption that the input data has n-th order cyclic symmetry and the strongly convex function φ is given in the form of (39) above. The configuration of the analysis apparatus according to the third embodiment is the same as that of the first embodiment, i.e., the same as that shown in FIG. 1, and the processing operation procedure is the same as that shown in FIG.

[0079] The individual components will be described below. The cyclic symmetric dual problem solving unit 11 in the third embodiment uses two vectors α and β and a cost matrix, which are part of the input data a, b, and c, as input data for the dual problem expressed by the above (22) (S11), solves the dual problem, and outputs an optimal solution of the dual problem expressed by the following (40) (S12).

[0080]

[0081] However, in the third embodiment, the strongly convex function φ in the optimal transportation problem with strongly convex regularization terms as shown in the above (11) is specifically given in the form of the above (39). Therefore, when the alternating minimization algorithm disclosed in Non-Patent Document 3 is used as an algorithm for solving the dual problem as shown in the above (22), the above (26) can be specifically expressed as the following (41).

[0082]

[0083] Also, K in (41) above ｉｊ can be expressed as follows (42):

[0084]

[0085] The w in (41) above ｉ is optimal when the value obtained by the above formula (41) is 0, so this optimal w ｉ is given in a closed form as shown in (43) below.

[0086]

[0087] Similarly, the optimal z ｊ is given in a closed form as shown in (44) below.

[0088]

[0089] Here, by defining an equation such as the following (45), the above (43) and (44) can be expressed as an arithmetic equation for matrix multiplication such as the following (46).

[0090]

[0091] After all, the cyclic symmetric dual problem solving unit 11 in the third embodiment solves the above q ｊ is initialized to an appropriate value, for example, a vector with all elements set to 1, and the matrix multiplication expressed by (46) above is repeated to obtain the optimal solution p ＊ , q ＊ can be calculated and output.

[0092] Next, the optimal transportation problem solving unit 12 in the third embodiment first calculates the optimal solution p ＊ , q ＊ is used to find the value of the left side of the relational expression shown in (47) below.

[0093]

[0094] The value on the left side of the above (47) is defined as shown in the following (48).

[0095]

[0096] Then, the optimal transportation problem solving unit 12 calculates the optimal transportation matrix T ＊The optimal transportation problem solving unit 12 can calculate the optimal transportation distance, which is the optimal objective function value of the optimal transportation problem with a strongly convex regularization term as shown in (11) above, from (31) above, and calculate this value and the optimal transportation matrix T ＊ is output (S13).

[0097] Next, a fourth embodiment will be described. This third embodiment is an embodiment in which the strongly convex function φ in the optimal transportation problem with strongly convex regularization terms as shown in (11) above in the second embodiment is given in the form of (39) above.

[0098] In other words, the fourth embodiment describes an algorithm that can quickly solve the entropy regularized optimal transportation problem, which is an optimal transportation problem when the input data of the strongly convex optimal transportation problem with entropy regularization as shown in (11) above approximately satisfies the assumption that the input data has n-th order cyclic symmetry. The configuration of the analysis apparatus according to the fourth embodiment is the same as that of the second embodiment, i.e., the same as that shown in FIG. 3, and the processing operation flow procedure is the same as that shown in FIG.

[0099] The individual components will be described below. The cyclic symmetric dual problem solving unit 11a in the fourth embodiment calculates two vectors α and β from input data a and b using the above (33), and based on the results of this calculation, performs the same processing as the cyclic symmetric dual problem solving unit 11 in the third embodiment to obtain the optimal solution p ＊ , q ＊ is calculated and output (S21, S22).

[0100] Next, the original dual problem solving unit 13a of the fourth embodiment considers the dual problem of the above (34) in the same way as in the second embodiment. However, in the fourth embodiment, the strongly convex function φ in the optimal transportation problem with strongly convex regularization terms as shown in the above (11) is specifically given in the form of the above (39), and the above (35) in the second embodiment can be specifically expressed in the form of the following (49) in the fourth embodiment, and K of the above (49) ｉｊ can be expressed as follows (50).

[0101]

[0102] f in (49) above ｉ is optimal when the value obtained by the formula (49) is 0, so this optimal f ｉ is given in a closed form as shown in (51) below.

[0103]

[0104] Similarly, the optimal g ｉ is given in closed form as shown in (52) below.

[0105]

[0106] Here, by defining an equation such as the following (53), the above (51) and (52) can be expressed as an arithmetic equation for matrix multiplication such as the following (54).

[0107]

[0108] Ultimately, the original dual problem solving unit 13a in the fourth embodiment solves v in (54) above. ｊ is initialized appropriately, for example, to a vector with all elements set to 1, and the matrix multiplication expressed by (54) above is repeated to obtain the optimal solution u ＊ , v ＊ Here, when repeating the matrix multiplication calculation of (54) for u and v, the original dual problem solving unit 13a calculates and outputs the optimal solution p ＊ , q ＊ is used to obtain the following (55).

[0109]

[0110] In this way, in the fourth embodiment, the alternating minimization algorithm is started from an initial solution that takes into account the ideal cyclic symmetry of the input data, rather than from an arbitrary initial solution. This makes it possible to efficiently find an optimal solution to the optimal transportation problem with a strongly convex regularization term, such as that shown in (11) above, where the input data has approximate cyclic symmetry.

[0111] Next, the optimal transportation problem solving unit 12a first calculates the dual problem optimal solution u as expressed in (34) above, which is the output from the cyclic symmetric dual problem solving unit 11a. ＊ , v ＊ is used to find the value of the left side of the relational expression shown in (56) below.

[0112]

[0113] The value on the left side of the above (47) is defined as shown in the following (57).

[0114]

[0115] In addition, the optimal transportation problem solving unit 12a can calculate the optimal transportation distance, which is the optimal objective function value of the optimal transportation problem with a strongly convex regularization term as shown in (11) above, using (31) above, and calculate this value and the optimal transportation matrix T ＊ is output (S24).

[0116] Fig. 5 is a block diagram showing an example of the hardware configuration of an analytical device according to one embodiment of the present invention. In the example shown in Fig. 5, the analytical device 100 according to the first embodiment is configured, for example, by a server computer or a personal computer, and has a hardware processor 111A such as a CPU (Central Processing Unit). A program memory 111B, a data memory 112, an input / output interface 113, and a communication interface 114 are connected to this hardware processor 111A via a bus 115. The same applies to the analytical device 100a according to the second embodiment.

[0117] The communication interface 114 includes, for example, one or more wireless communication interface units, and enables transmission and reception of information to and from a communication network NW. As the wireless interface, for example, an interface that adopts a low-power wireless data communication standard such as a wireless LAN (Local Area Network) is used.

[0118] An input device 300 and an output device 400, which are attached to the analysis apparatus 100 and used by a user or the like, are connected to the input / output interface 113. The input / output interface 113 receives operation data input by a user or the like through the input device 300, such as a keyboard, touch panel, touchpad, or mouse, and outputs output data to an output device 400, which may include a display device using a liquid crystal or organic electroluminescence (EL) display, for display. The input device 300 and the output device 400 may be devices built into the analysis apparatus 100, or may be input devices and output devices of other information terminals that can communicate with the analysis apparatus 100 via the network NW.

[0119] The program memory 111B is a non-transitory tangible storage medium that is a combination of a non-volatile memory that can be written to and read from at any time, such as a hard disk drive (HDD) or a solid state drive (SSD), and a non-volatile memory such as a read only memory (ROM), and stores programs necessary to execute various control processes, etc., according to one embodiment.

[0120] The data memory 112 is a tangible storage medium that is, for example, a combination of the above-mentioned nonvolatile memory and a volatile memory such as RAM (Random Access Memory), and is used to store various data acquired and created during various processes performed by the analysis device 100.

[0121] The analytical device 100 according to one embodiment of the present invention may be configured as a data processing device having software-based processing function units. Storage areas used as work memory or the like by each unit of the analytical device 100 may be configured using the data memory 112 shown in FIG. 5. However, these configured storage areas are not essential components within the analytical device 100, and may be areas provided in, for example, an external storage medium such as a USB (Universal Serial Bus) memory, or a storage device such as a database server located in the cloud.

[0122] The processing function unit can be realized by having the hardware processor 111A read and execute a program stored in the program memory 111B, but the processing function unit may also be realized in various other forms, including an integrated circuit such as an application specific integrated circuit (ASIC) or a field-programmable gate array (FPGA).

[0123] The methods described in each embodiment may be stored as a program (software means) that can be executed by a computer on a recording medium such as a magnetic disk (e.g., a floppy disk, a hard disk, etc.), an optical disk (e.g., a CD-ROM, a DVD, an MO, etc.), or a semiconductor memory (e.g., a read-only memory (ROM), a random access memory (RAM), a flash memory, etc.), or may be transmitted and distributed via a communication medium. The program stored on the medium also includes a configuration program that configures the software means (including not only execution programs but also tables and data structures) that the computer executes. The computer that realizes this device reads the program stored on the recording medium and, in some cases, configures the software means using the configuration program, and executes the above-mentioned processing by having the operation controlled by this software means. The term "recording medium" as used herein is not limited to a storage medium for distribution, but also includes a storage medium such as a magnetic disk or semiconductor memory installed inside the computer or in a device connected via a network.

[0124] The present invention is not limited to the above-described embodiments, and various modifications can be made in the implementation stage without departing from the spirit of the invention. Furthermore, the embodiments may be implemented in appropriate combinations, in which case the combined effects can be obtained. Furthermore, the above-described embodiments include various inventions, and various inventions can be extracted by combining selected elements from the disclosed elements. For example, if the problem can be solved and the desired effect can be obtained even if some elements are deleted from all elements shown in the embodiments, the configuration from which these elements are deleted can be extracted as an invention.

[0125] 100, 100a...Analysis device 11, 11a...Cyclic symmetric dual problem solving unit 12, 12a...Optimal transportation problem solving unit 13a...Original dual problem solving unit

Claims

1. An analysis device comprising: a cyclic symmetric dual problem solving unit that inputs first and second vectors which are part of symmetric components of first and second input data which have cyclic symmetry, and a cost matrix which is part of symmetric components of third input data which also has cyclic symmetry, into a dual problem based on an optimal transportation problem with a strongly convex regularization term, and outputs an optimal solution to the dual problem; and an optimal transportation problem solving unit that uses the optimal solution output by the cyclic symmetric dual problem solving unit to find and output an optimal solution to the optimal transportation problem with strongly convex regularization term and an optimal objective function value.

2. An analysis device comprising: a cyclic symmetric dual problem solving unit that calculates first and second vectors from first and second input data having approximate cyclic symmetry, inputs the calculated first and second vectors and a cost matrix that is a part of the symmetric components of a third input data into a first dual problem based on an optimal transportation problem with a strongly convex regularization term, and outputs an optimal solution of the first dual problem; an original dual problem solving unit that converts the optimal solution output by the cyclic symmetric dual problem solving unit into an initial solution that assumes an ideal state of cyclic symmetry of the input data, and outputs an optimal solution of a second dual problem based on the optimal transportation problem with a strongly convex regularization term using the initial solution; and an optimal transportation problem solving unit that uses the optimal solution output by the original dual problem solving unit to find and output an optimal solution of the optimal transportation problem with a strongly convex regularization term and an optimal objective function value.

3. The analysis device described in claim 1, wherein the optimal transportation problem with strong convex regularization term is an optimal transportation problem with entropy regularization term including a strongly convex function with defined conditions for possible values, the cyclic symmetric dual problem solving unit inputs first and second vectors which are part of the symmetric components of the first and second input data and a cost matrix which is part of the symmetric components of the third input data to a dual problem based on the optimal transportation problem with entropy regularization term and outputs an optimal solution to the dual problem, and the optimal transportation problem solving unit uses the optimal solution output by the cyclic symmetric dual problem solving unit to determine and output an optimal solution to the optimal transportation problem with entropy regularization term and an optimal objective function value.

4. The analysis device of claim 2, wherein the optimal transportation problem with strong convex regularization term is an optimal transportation problem with entropy regularization term including a strongly convex function whose possible values ​​are determined; the cyclic symmetric dual problem solving unit calculates the first and second vectors from the first and second input data, and inputs the calculated first and second vectors and a cost matrix that is a part of the symmetric components of the third input data into the first dual problem to output an optimal solution of the first dual problem; the original dual problem solving unit converts the optimal solution output by the cyclic symmetric dual problem solving unit into the initial solution, and uses the initial solution to output an optimal solution of a second dual problem based on the optimal transportation problem with entropy regularization term; and the optimal transportation problem solving unit uses the optimal solution output by the original dual problem solving unit to determine and output an optimal solution of the optimal transportation problem with entropy regularization term and an optimal objective function value.

5. A method performed by an analysis device, comprising: a cyclic symmetric dual problem solving unit of the analysis device calculating first and second vectors from first and second input data having approximate cyclic symmetry, inputting the calculated first and second vectors and a cost matrix that is a part of the symmetric components of the third input data into a first dual problem based on an optimal transportation problem with a strongly convex regularization term, and outputting an optimal solution of the first dual problem; an original dual problem solving unit of the analysis device converting the optimal solution output by the cyclic symmetric dual problem solving unit into an initial solution that assumes an ideal state of cyclic symmetry of the input data, and outputting an optimal solution of the second dual problem based on the optimal transportation problem with a strongly convex regularization term using the initial solution; and an optimal transportation problem solving unit of the analysis device using the optimal solution output by the original dual problem solving unit to determine and output an optimal solution of the optimal transportation problem with a strongly convex regularization term and an optimal objective function value.

6. An analytical processing program that causes a processor to function as each part of the analytical device according to any one of claims 1 to 4.

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