Solution apparatus, solution method, and program
The solution-finding device and method enhance DC optimization by using a specific configuration of convex functions to address perturbation loss and amplitude underestimation, ensuring accurate and stable solutions to underdetermined inverse problems.
Patent Information
- Application Number
- JP2024100820
- Authority / Receiving Office
- JP · JP
- Patent Type
- Applications
- Current Assignee / Owner
- Filing Date
- 2024-06-21
- Publication Date
- 2026-01-08
AI Technical Summary
Existing optimization techniques for underdetermined inverse problems, such as DC optimization and sparse regularization, suffer from perturbation loss and amplitude underestimation, particularly when using non-convex functions like ROWL, which can hinder calculation processes.
A solution-finding method and device that employs DC optimization with a specific configuration of convex functions, including an L1-convex function and an inner product of non-decreasing vectors, to minimize perturbation loss and amplitude underestimation, utilizing a control unit to execute the process.
The method effectively suppresses perturbation loss and amplitude underestimation in DC optimization, ensuring stable calculation processes and accurate solutions to underdetermined inverse problems.
Smart Images
Figure 2026002669000001_ABST
Abstract
Description
[Technical Field]
[0001] The present invention relates to a solution-finding device, a solution-finding method, and a program. [Background technology]
[0002] Underdetermined inverse problems are a technology required in many areas, such as sensing and computational imaging. As a result, research into them is active, and various regularization techniques and solution algorithms have been proposed. [Prior art documents] [Non-patent literature]
[0003] [Non-Patent Document 1] T. Sasaki, Y. Bandoh, and M. Kitahara, “Sparse Regularization Based on Reverse Ordered Weighted L1-Norm and Its Application to Edge-Preserving Smoothing,” in ICASSP 2024 - 2024 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), IEEE, Apr. 2024, pp. 9531-9535. Summary of the Invention [Problem to be solved by the invention]
[0004] Solutions to such inverse problems can be obtained through optimization, but optimization is performed on a regularized function representing the inverse problem to improve estimation accuracy and generalization performance. However, regularization can cause perturbation loss and amplitude underestimation. Perturbation loss and amplitude underestimation occur particularly in the widely used sparse regularization. Note that perturbation loss refers to the phenomenon in which the magnitude of the estimated signal, which is an estimated signal when the magnitude of the original signal is small, becomes almost zero, and amplitude underestimation refers to the phenomenon in which the difference between the original signal and the estimated signal is large.
[0005] Therefore, Non-Patent Document 1 proposed ROWL (Reverse Ordered Weighted L1-Norm) as a regularization technique. ROWL is a regularization technique that suppresses the occurrence of perturbation disappearance and amplitude underestimation more than sparse regularization. However, because ROWL is a non-convex function, there are limitations on the optimization techniques that can be applied. Note that the condition for applicability here is that even if ROWL is applied, it does not make it impossible to perform calculations according to the algorithm, and a solution that is guaranteed to suppress the occurrence of perturbation disappearance and amplitude underestimation can be obtained.
[0006] There are various optimization techniques for solving underdetermined inverse problems, but one widely used technique is DC (Difference of Convex Function) optimization. DC optimization minimizes the difference between two convex functions. DC optimization also suffers from perturbation loss and amplitude underestimation due to sparse regularization. Therefore, applying ROWL would reduce the occurrence of perturbation loss and amplitude underestimation. However, because DC optimization minimizes the difference between two convex functions, using ROWL, a non-convex function, for one of the two convex functions not only fails to reduce the occurrence of perturbation loss and amplitude underestimation compared to sparse regularization, but also may make it impossible to perform calculations according to the algorithm in the first place.
[0007] In view of the above circumstances, an object of the present invention is to provide a technique for further suppressing the occurrence of perturbation loss and amplitude underestimation in DC optimization. [Means for solving the problem]
[0008] One aspect of the present invention is the optimization of Difference of Convex Functions (DC), which is an optimization that minimizes the difference between two convex functions. and a control unit that executes a solution-finding process to obtain a solution to an underdetermined inverse problem by DC optimization, wherein one of the two convex functions is a sum of an L1-convex function and a function representing the inverse problem, and the other of the two convex functions is an inner product of a first non-decreasing vector and an absolute value vector, the L1-convex function being a product of a maximum value among the element values of a second non-decreasing vector, which is a predetermined vector of non-negative elements and whose element values change monotonically and non-decreasingly from smaller element numbers to larger element numbers, and the L1-norm of a variable vector, which is a vector whose elements are optimization variables, the first non-decreasing vector being a result of subtracting the second non-decreasing vector from a vector whose element values are the maximum values among the element values of the second non-decreasing vector, and the absolute value vector is a vector that satisfies a condition that the value of the element with the mth smallest element number (m is an integer equal to or greater than 1) has the mth largest absolute value among the element values of the variable vector, and the solution of the inverse problem is a solution obtained by the DC optimization.
[0009] One aspect of the present invention includes a control unit that executes a solution-finding process to obtain a solution to an underdetermined inverse problem by DC (Difference of Convex Function) optimization, which is optimization that minimizes the difference between two convex functions, wherein one of the two convex functions is a sum of an L1 convex function and a function that represents the inverse problem, and the other of the two convex functions is an inner product of a first non-decreasing vector and an absolute value vector, and the L1 convex function is a predetermined vector of non-negative elements, and element values of a second non-decreasing vector change monotonically non-decreasingly from smaller element numbers to larger element numbers, and the L1 norm of a variable vector, which is a vector whose elements are variables of optimization. a product of the first non-decreasing vector and the second non-decreasing vector, wherein the first non-decreasing vector is the result of subtracting the second non-decreasing vector from a vector whose element values are the maximum of the element values of the second non-decreasing vector; the absolute value vector is a vector that satisfies the condition that the value of the element with the mth smallest element number (m is an integer greater than or equal to 1) has the mth largest absolute value among the element values of the variable vector; and the solution to the inverse problem is the solution obtained by the DC optimization. This is a solution-finding method executed by a solution-finding device, and includes a control step of executing the solution-finding process.
[0010] One aspect of the present invention is a program for causing a computer to function as the above-described solution-finding device. [Effects of the Invention]
[0011] The present invention makes it possible to further suppress the occurrence of perturbation loss and amplitude underestimation in DC optimization. [Brief explanation of the drawings]
[0012] [Figure 1] FIG. 1 is an explanatory diagram illustrating a solution-finding apparatus according to an embodiment. [Figure 2] FIG. 2 is a diagram showing an example of the hardware configuration of a solution solving apparatus according to an embodiment. [Figure 3] 3 is a flowchart showing an example of a flow of processing executed by the solution finding device in the embodiment. DETAILED DESCRIPTION OF THE INVENTION
[0013] (Embodiment) 1 is an explanatory diagram illustrating a solution-finding device 1 according to an embodiment. The solution-finding device 1 includes a control unit 11 including a processor 91, such as a CPU (Central Processing Unit), a GPU (Graphics Processing Unit), or an NPU (Neural Network Processing Unit), and a memory 92, which are connected via a bus.
[0014] The control unit 11 executes, for example, a solution-finding process. The solution-finding process is a process for obtaining a solution to a problem to be solved (hereinafter referred to as the "problem to be solved") by DC (Difference of Convex function) optimization. The solution to the problem to be solved is the solution obtained by this DC optimization. Specifically, the problem to be solved is an underdetermined inverse problem.
[0015] As is well known, DC optimization is an optimization that minimizes the difference between two convex functions. Here, we will explain what specific convex functions the two convex functions in DC optimization executed in the solution-finding process are. For simplicity of explanation, one of the two convex functions in DC optimization executed in the solution-finding process will be referred to as the first convex function, and the other will be referred to as the second convex function.
[0016] The first convex function is the sum of an L1 convex function and a function that represents the problem to be solved. The second convex function is the inner product of a first non-decreasing vector and an absolute value vector.
[0017] The L1 convex function is the product of the maximum value of the elements of the second non-decreasing vector and the L1 norm of the variable vector. The second non-decreasing vector is a predetermined vector of non-negative elements, in which the element values change monotonically and non-decreasingly from the smallest element number to the largest element number. The variable vector is a vector whose elements are optimization variables. Note that the optimization variables are optimization variables in the problem to be solved. Therefore, the optimization variables here are optimization variables in the DC optimization performed in the solution process.
[0018] The first non-decreasing vector is the result of subtracting the second non-decreasing vector from the maximum value vector. The maximum value vector is a vector whose element value is the maximum value of the second non-decreasing vector. In other words, it is the result of multiplying a vector whose element values are all 1 by the maximum value of the second non-decreasing vector.
[0019] An absolute vector is a vector that satisfies the condition that the value of the element with the mth smallest element number (m is an integer equal to or greater than 1) indicates the mth largest absolute value among the element values of the variable vector.
[0020] <Explanation using mathematical formulas for solution-finding process> The solution process will be explained using mathematical expressions. The expression of DC decomposition of the DC function in DC optimization executed in the solution process is expressed by the following equation (1).
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[0022] Here, x is a variable vector. Therefore, each element of the variable vector x is an optimization variable. The variable vector x can be expressed by the following formula (2) if each element is clearly stated. Note that N is a predetermined integer equal to or greater than 1, and n is an integer equal to or greater than 1 and equal to or less than N.
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[0024] <<First term of equation (1)>> We will now explain the first term in equation (1). That is, we will explain the function f(x). The function f(x) is a convex function that represents the problem to be solved. The function f(x) is a function with the variable vector x as the independent variable. Since the function f(x) represents the problem to be solved, it is a function expressed, for example, by the following equation (3). y and e are vectors, and A is a matrix. Here, vector y represents known information. Known information is, for example, information obtained by measurement. Vector e is also known information. Vector e represents, for example, noise. Variable vector x represents unknown information.
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[0026] The forward problem corresponding to the inverse problem expressed by equation (3) is expressed by the following equation (4).
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[0028] Note that the vector e in the second term on the right-hand side of equation (4) does not necessarily have to exist in the forward problem. If the vector e does not exist in the forward problem, the vector e also does not exist in the inverse problem. Therefore, the vector e does not necessarily have to be included in the function f(x). Note that the matrix A may be either a square matrix or a non-square matrix.
[0029] <<The second term of equation (1)>> The function of the second term in equation (1) will be explained. In other words, the function of the second term in equation (1) is a function of the second term in the parentheses of equation (1). The second term in equation (1) is the value w N and the L1 norm of the variable vector x. n is the maximum value among the values of the elements of the vector w expressed by the following equation (5).
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[0032] Each element w1~w of vector w N is a predetermined value. In this way, the vector w is a predetermined vector of non-negative elements, and the element values change monotonically non-decreasingly from elements with smaller element numbers to elements with larger element numbers. In other words, the vector w is a second non-decreasing vector.
[0033] Therefore, the function in the second term of equation (1) is an L1 convex function. Therefore, the function expressed by the following equation (7) in equation (1) (i.e., the function in parentheses in equation (1)) is a L1 convex function.
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[0035] <<The third term of equation (1)>> The symbol of the third term in equation (1) will be explained. The symbol of the third term in equation (1) is defined by the following equation (8). That is, the symbol of the third term in equation (1) is the vector ω and vector |x| defined by the following equation (9). ↓ Represents the dot product of the vector |x| ↓ is a vector that satisfies the condition that the value of the element with the smallest element number (m is an integer greater than or equal to 1) is the mth largest absolute value among the element values of the variable vector x. Therefore, the vector |x| ↓ is the absolute value vector.
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[0038] From the definition of equation (9), the elements of vector ω satisfy the relationship of equation (10) below.
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[0040] The vector expressed by the following equation (11) in equation (9) is a maximum value vector because the value of each element is the maximum value among the element values of vector w. Therefore, vector ω is the result of subtracting the second non-decreasing vector from the maximum value vector, so vector ω is the first non-decreasing vector.
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[0042] Thus, the vector ω is the first non-decreasing vector, and the vector |x| ↓ is an absolute value vector, and therefore, by the definition of equation (8), the third term in equation (1) is a second convex function.
[0043] << vector |x| ↓ Example>> Just to be safe, the vector |x| ↓ An example of the vector |x| ↓ is an absolute value vector, so when each element of the variable vector x satisfies the condition expressed by the following equation (12), for example, the vector |x| ↓ is expressed by the following equation (13).
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[0046] Note that the function in the second term of equation (1) is a function of the L1 norm, and is therefore a convex function that can be near-mapped. Therefore, a first convex function is a convex function, since it is the sum of a convex function and a convex function. Furthermore, the third term of equation (1) (i.e., a second convex function) is also a convex function that can be near-mapped (see Reference 1). From the above, the DC function in equation (1) is a differentiable convex function.
[0047] Reference 1: Xiangrong Zeng and Mario AT Figueiredo “Decreasing Weighted Sorted Regularization” IEEE SIGNAL PROCESSING LETTERS, VOL. 21, NO. 10, OCTOBER 2014, pp. 1240-1244.
[0048] Since the DC function is expressed by equation (1), the DC optimization performed in the solution process is expressed by the following equation (14): In other words, the solution process optimizes the following equation (14), and the value of the obtained variable vector x is the solution to the variable solution target problem.
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[0050] In fact, the function of the second term in formula (1) and the function of the third term in formula (1) satisfy the relationship expressed by the following formula (15) with ROWL (Reverse Ordered Weighted L1-Norm) described in Non-Patent Document 1. ROWL is a non-convex function (see Non-Patent Document 1). In formula (15), the symbol on the left side represents ROWL.
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[0052] The theorem that shows the relationship in the following equation (15) is called the Sasaki-Bando-Kitahara-Ono theorem.
[0053] <<Proof of the Sasaki-Bando-Kitahara-Ono Theorem>> We prove the Sasaki-Bando-Kitahara-Ono theorem. The definition of ROWL is given by the following equation (16).
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[0055] That is, ROWL is a vector w that is a second non-decreasing vector and a vector |x| that is an absolute value vector. ↓ By using equation (9) for equation (16), the following equation (17) is obtained. Note that vector 1 N is an N-dimensional vector with elements all equal to 1.
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[0057] Expanding the right-hand side of equation (17), the following relationship in equation (18) is obtained.
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[0059] Vector 1 N Considering the definition of the first term on the right side of equation (18), the value w N and the L1 norm of the variable vector x. Therefore, the first term on the right side of equation (18) is an L1 convex function. And, from equation (8), the second term on the right side of equation (18) is a second-order convex function. Therefore, the relationship in equation (19) below is obtained.
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[0061] Equation (15) can be obtained from equations (16) to (19). QED
[0062] <Effects of solution-finding processing> As described above, DC optimization is optimization that minimizes the difference between two convex functions. Meanwhile, ROWL is a regularization technique as described in Non-Patent Document 1, and is known to suppress the occurrence of perturbation loss and amplitude underestimation as described in Non-Patent Document 1. Therefore, if ROWL can be combined with DC optimization in the form of a regularization term, it is possible to obtain a solution to the inverse problem while further suppressing the occurrence of perturbation loss and amplitude underestimation.
[0063] However, ROWL is a non-convex function. Therefore, even if one of the convex functions in DC optimization, which is a technique applied to a function expressed by two convex functions, is ROWL, it is impossible to perform calculations according to the algorithm. Therefore, it is natural for those skilled in the art to think that ROWL cannot be applied to DC optimization.
[0064] In contrast, solution processing is a technology that applies ROWL to DC optimization, which at first glance seems impossible. The reason for this is explained below. The DC function in solution processing is expressed as a first-convex function, which is a convex function, and a second-convex function, which is also a convex function. Furthermore, the relationship in equation (15) holds due to the Sasaki-Bando-Kitahara-Ono theorem, so equation (1) is actually transformed into the following equation (20).
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[0066] Equation (20) is an equation in which the function f(x) is regularized by ROWL. Therefore, the solution process incorporates ROWL and then obtains a solution by DC optimization. This is the reason. Therefore, by executing the solution process, the occurrence of perturbation disappearance and amplitude underestimation in DC optimization is further suppressed.
[0067] <Example of hardware configuration> 2 is a diagram showing an example of the hardware configuration of the solution-finding device 1 in the embodiment. The solution-finding device 1 has a control unit 11 and executes a program. By executing the program, the solution-finding device 1 functions as a device having the control unit 11, an interface unit 12, and a storage unit 13.
[0068] More specifically, the processor 91 reads out a program stored in the storage unit 13 and stores the read out program in the memory 92. When the processor 91 executes the program stored in the memory 92, the solution-finding device 1 functions as a device including the control unit 11, the interface unit 12, and the storage unit 13.
[0069] The control unit 11 controls the operation of each functional unit included in the solution-finding device 1. The control unit 11 executes, for example, a solution-finding process. The control unit 11 acquires, for example, information stored in the memory unit 13. Specifically, the process of acquiring information stored in the memory unit 13 is reading.
[0070] The interface unit 12 includes a communication interface for connecting the solution-finding device 1 to an external device. The interface unit 12 communicates with the external device via wire or wirelessly.
[0071] The external device is, for example, a device that transmits information used for execution by the control unit 11. In such a case, the interface unit 12 acquires the information used for execution by the control unit 11 from the device that transmits the information used for execution by the control unit 11. The device that transmits the information used for execution by the control unit 11 is, for example, a device that transmits information indicating a function that represents the problem to be solved (hereinafter referred to as "problem information to be solved"). Since the problem information to be solved is information that indicates a function that represents the problem to be solved, the problem information to be solved can also be said to be information that indicates the problem to be solved in the form of a function.
[0072] The interface unit 12 may be configured to include input devices such as a mouse, keyboard, touch panel, etc. The interface unit 12 may be configured as an interface that connects these input devices to the solution-finding device 1. In this way, the input device of the interface unit 12 accepts input of various information to the solution-finding device 1 via wired or wireless connections. Note that the information does not necessarily have to be input to the communication interface of the interface unit 12, but may also be input to the input device of the interface unit 12. Therefore, information about the problem to be solved may be input to the input device of the interface unit 12, for example.
[0073] The interface unit 12 outputs, for example, various types of information. The interface unit 12 includes a display device such as a CRT (Cathode Ray Tube) display, a liquid crystal display, or an organic EL (Electro-Luminescence) display, and a speaker. The interface unit 12 may be configured as an interface that connects these display devices or speakers to the solution-finding device 1. Therefore, the display device and speaker included in the interface unit 12 output, for example, information acquired by a communication interface of the interface unit 12 or information input to an input device of the interface unit 12 as an image or sound.
[0074] The storage unit 13 is configured using a computer-readable storage medium (non-transitory computer-readable recording medium) such as a magnetic hard disk drive or a semiconductor storage device. The storage unit 13 stores various information related to the solution-finding device 1. The storage unit 13 stores various information generated by the operation of the control unit 11, for example. The storage unit 13 stores information used in the solution-finding process and information generated by the execution of the solution-finding process, for example. Therefore, the storage unit 13 may store information indicating the second non-decreasing vector in advance, for example. The storage unit 13 may exist on a cloud, for example.
[0075] 3 is a flowchart showing an example of the flow of processing executed by the solution-finding device 1 in the embodiment. The control unit 11 acquires solution-finding target problem information (step S101). Next, the control unit 11 executes solution-finding processing (step S102). By executing the solution-finding processing, a solution to the solution-finding target problem, which is indicated in the form of a function by the solution-finding target problem information obtained in step S101, is obtained.
[0076] The solution-finding device 1 configured in this way executes the solution-finding process, which makes it possible to further suppress the occurrence of perturbation loss and amplitude underestimation in DC optimization, as described in <Effects of the solution-finding process>.
[0077] (Variation) There are various algorithms for performing DC optimization, but among them, an algorithm that is guaranteed to converge to a stationary point may be used as the DC optimization algorithm. The solution process for performing DC optimization using such an algorithm not only further suppresses the occurrence of perturbation loss and amplitude underestimation, but also obtains a solution that is guaranteed to converge to a stationary point.
[0078] The algorithm used in DC optimization that is guaranteed to converge to a stationary point may be, for example, the general double-proximal gradient algorithm. Specifically, in the solution process for DC optimization using the general double-proximal gradient algorithm, alternating optimization of the following equations (21) and (22) is performed. In this alternating optimization, for example, the following equations (23) to (26) may be used. Note that equation (23) is an equation obtained by Moreau decomposition. Note that γ1 is defined as a positive step size, and γ2 is defined as a positive step size.
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[0085] Note that prox represents a proximity mapping defined by the following equation (27): In equation (27), η represents a positive real number, and the function U represents a mapping from an N-dimensional real number space to a one-dimensional real number space.
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[0087] The proximity map expressed by equation (27) is called the proximity map of the function U.
[0088] <Application example> An application example of the solution-finding process will be described. The solution-finding process may be used, for example, in super-resolution technology. When used in super-resolution, the function f(x) is specifically expressed by the following equation (28). In this case, the first convex function is specifically expressed by the following equation (29), and the second convex function is specifically expressed by the following equation (30).
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[0092] Here, A is a composite linear mapping of, for example, sampling, low-pass filtering, and inverse discrete cosine transform.
[0093] The solution-finding device 1 may be implemented using a plurality of information processing devices communicably connected via a network. In this case, the functional units of the solution-finding device 1 may be distributed and implemented among the plurality of information processing devices.
[0094] All or part of the functions of the solution-finding device 1 may be realized using hardware such as an ASIC (Application Specific Integrated Circuit), a PLD (Programmable Logic Device), or an FPGA (Field Programmable Gate Array). The program may be recorded on a computer-readable recording medium. Examples of computer-readable recording media include portable media such as flexible disks, magneto-optical disks, ROMs, and CD-ROMs, and storage devices such as hard disks built into computer systems. The program may be transmitted via a telecommunications line.
[0095] Although an embodiment of the present invention has been described above in detail with reference to the drawings, the specific configuration is not limited to this embodiment, and includes designs within the scope of the gist of the present invention. [Explanation of symbols]
[0096] 1...solution-finding device, 11...control unit, 12...interface unit, 13...storage unit, 91...processor, 92...memory
Claims
1. a control unit that executes a solution process to obtain a solution to an underdetermined inverse problem by DC (Difference of Convex Function) optimization, which is an optimization that minimizes the difference between two convex functions; Equipped with one of the two convex functions is a sum of an L1 convex function and a function representing the inverse problem; the other of the two convex functions is the dot product of the first non-decreasing vector and the absolute value vector; the L1 convex function is a product of the maximum value of the element values of a second non-decreasing vector, which is a predetermined vector of non-negative elements and whose element values change monotonically non-decreasingly from smaller element numbers to larger element numbers, and the L1 norm of a variable vector, which is a vector whose elements are optimization variables; the first non-decreasing vector is a result of subtracting the second non-decreasing vector from a vector whose element values are the maximum values of the elements of the second non-decreasing vector; The absolute value vector is a vector that satisfies the condition that the value of the element having the mth smallest element number (m is an integer of 1 or more) indicates the mth largest absolute value among the element values of the variable vector. The solution to the inverse problem is the solution obtained by the DC optimization. Solving device.
2. The DC optimization algorithm is an algorithm that is guaranteed to converge to a stationary point. The solution-finding apparatus according to claim 1 .
3. The algorithm used in the DC optimization, which is guaranteed to converge to a stationary point, is the General Double-Proximal Gradient Algorithm. The solution-finding apparatus according to claim 2 .
4. DC (Difference of Convex Functions) is an optimization that minimizes the difference between two convex functions. a control unit that executes a solution-finding process to obtain a solution to an underdetermined inverse problem by DC optimization, wherein one of the two convex functions is a sum of an L1 convex function and a function representing the inverse problem, and the other of the two convex functions is an inner product of a first non-decreasing vector and an absolute value vector, the L1 convex function being a product of a maximum value among element values of a second non-decreasing vector, the second non-decreasing vector being a predetermined vector of non-negative elements, element values of which change monotonically non-decreasingly from smaller element numbers to larger element numbers, and the L1 norm of a variable vector, the vector having optimization variables as elements, the first non-decreasing vector being a result of subtracting the second non-decreasing vector from a vector whose element values are the maximum values among element values of the second non-decreasing vector, and the absolute value vector is a vector that satisfies a condition that the value of the element with the mth smallest element number (m is an integer equal to or greater than 1) has an absolute value that is the mth largest among element values of the variable vector, and the solution of the inverse problem is a solution obtained by the DC optimization, a control step for executing the solution-finding process; A solution method having
5. A program for causing a computer to function as the solution-finding device according to any one of claims 1 to 3.