Information processing apparatus, information processing method, and program

Abstract

To calculate solution with proper sparsity for a problem requiring solution from observation information.SOLUTION: An information processing apparatus according to the present invention has a storage unit for storing model information representing relationship between a plurality of first variables and a plurality of second variables. The storage unit stores first evaluation function information for use in evaluating first solution including respective values of the plurality of second variables calculated based on the model information and second evaluation function information for use in evaluating parameters included in the first evaluation function information. A calculation unit, based on the model information, the first evaluation function information, and the second evaluation function information, calculates second solution including the respective values of the plurality of second variables from observation information including respective values of the plurality of first variables. The first estimation function information includes first evaluation information representing accuracy of the first solution and second evaluation information representing sparsity of the first solution. The parameter represents a degree of influence of the second evaluation information on an evaluation value of the first solution which is calculated from the first solution based on the first evaluation function information. The calculating unit determines values of the parameters based on the second evaluation function information and calculates the second solution based on the determined values.SELECTED DRAWING: Figure 1

Description

[Technical field]

[0001] The present invention relates to information processing. [Background technology]

[0002] In recent years, large-scale data has been acquired, and attempts have been made to extract important information from the acquired data. For example, in ground-penetrating radar exploration, a measurement image is generated from a measured signal.

[0003] However, the generated measurement image may be unclear due to unnecessary signals such as noise. In addition, because the measurement image does not directly represent the underground structure, specialized knowledge or experience may be required to determine whether the target has been detected. In such underground radar exploration, estimating the distribution of the reflection coefficient of the underground structure may clarify the underground structure and improve its interpretability.

[0004] 2. Description of the Related Art Regarding underground exploration, a reflected wave identification method is known for identifying a signal of a reflected wave of a sound wave emitted with a known waveform (see, for example, Patent Document 1). [Prior art documents] [Patent documents]

[0005] [Patent Document 1] Patent No. 6778572 Summary of the Invention [Problem to be solved by the invention]

[0006] Subsurface radar exploration is an example of geophysical exploration, and is an exploration method that visualizes the underground structure by imaging the transmitted and received radio wave signals. The received signal obtained by measurement is expressed as the convolution of the transmitted signal E0 and the reflection coefficient distribution r. In addition, since the actual measurement includes the observation noise n, the received signal Eobs is expressed by the following equation.

[0007] Eobs=r*E0+n (1)

[0008] r*E0 represents the convolution of the transmission signal E0 and the reflection coefficient distribution r. If the reflection coefficient distribution r in equation (1) can be sparsely estimated, the reflection boundary in the ground will become clearer, making it easier to determine the presence or absence of buried objects.

[0009] L1 regularization is an example of an effective method for extracting important information from a large amount of information. Regularization is one of the methods used to prevent overfitting, and it can minimize the number of explanatory variables while maintaining the accuracy of the model.

[0010] LASSO (Least Absolute Shrinkage and Selection Operator) is an example of regression analysis using L1 regularization, and is used in the fields of signal processing, machine learning, etc. With LASSO, even when observation noise n is added, it may be possible to restore a sparse reflection coefficient distribution r without applying noise reduction processing such as filters.

[0011] However, when applying LASSO to the problem of estimating the reflection coefficient distribution r, it is difficult to select an appropriate regularization parameter for sparsely estimating the reflection coefficient distribution r.

[0012] This problem is not limited to the case of applying LASSO to estimate the distribution of reflection coefficients in underground radar exploration, but occurs when obtaining sparse solutions from observational information in various problems.

[0013] In one aspect, the present invention aims to obtain a solution having appropriate sparsity for a problem of obtaining a solution from observed information. [Means for solving the problem]

[0014] According to one embodiment, an information processing device includes a storage unit and a calculation unit. The storage unit stores model information representing a relationship between a plurality of first variables and a plurality of second variables. The storage unit further stores first evaluation function information for evaluating a first solution including values ​​of the plurality of second variables calculated based on the model information, and second evaluation function information for evaluating parameters included in the first evaluation function information.

[0015] The calculation unit determines a second solution including values ​​of each of a plurality of second variables from observation information including values ​​of each of a plurality of first variables based on the model information, the first evaluation function information, and the second evaluation function information.

[0016] The first evaluation function information includes first evaluation information representing the accuracy of the first solution and second evaluation information representing the sparsity of the first solution. The parameter represents the influence of the second evaluation information on the evaluation value of the first solution obtained from the first solution based on the first evaluation function information. The calculation unit determines the value of the parameter based on the second evaluation function information, and obtains the second solution based on the determined value of the parameter. Effect of the Invention

[0017] In one aspect, it is possible to obtain a solution having appropriate sparsity for a problem that requires a solution from observed information. [Brief description of the drawings]

[0018] [Figure 1] FIG. 2 is a functional configuration diagram of the information processing apparatus according to the embodiment. [Diagram 2] 13 is a flowchart of a calculation process. [Diagram 3] FIG. 2 is a functional configuration diagram showing a specific example of an information processing device. [Figure 4] FIG. 13 is a diagram showing observation information in estimation of a reflection coefficient distribution. [Diagram 5] FIG. 1 is a diagram showing the distribution of reflection coefficients of the ground. [Figure 6] 11 is a flowchart of a first optimum solution estimation process. [Figure 7A]13 is a flowchart (part 1) of a second optimal solution estimation process. [Figure 7B] 13 is a flowchart (part 2) of the second optimum solution estimation process. [Figure 8] FIG. 11 is a diagram showing a first estimation result of a reflection coefficient distribution. [Figure 9] FIG. 11 is a diagram showing a second estimation result of the reflection coefficient distribution. [Figure 10] FIG. 2 is a hardware configuration diagram of an information processing device. DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0019] Hereinafter, the embodiments will be described in detail with reference to the drawings.

[0020] The observation model in the problem of estimating unknown quantities from observed information can be described by a linear transformation such as the following equation.

[0021] y=Φx (2)

[0022] y is a vector representing observed information, and x is a vector representing unknown quantities. Φ is an observation matrix representing a linear transformation. When LASSO is applied to this problem, for example, a solution x that minimizes an evaluation function F1 as shown in the following equation is obtained as the optimal solution.

[0023] F1=(1 / 2)(L2(Φx-y)) 2 +λ(L1(x)) (3)

[0024] L2() denotes the L2 norm of a vector, L1() denotes the L1 norm of a vector, and λ is a positive real number that represents a regularization parameter.

[0025] The sparsity of a solution x estimated by LASSO varies more significantly depending on the value of λ, and the larger λ is, the more zeros there are in the components of x. Therefore, a smaller λ results in a more overfitted solution x, and a larger λ results in a sparser solution x. However, there is a trade-off between the error of the solution x and the sparsity of the solution x, and if the value of λ is selected by considering only the error, the solution x may become overfitted and lose its sparsity.

[0026] Fig. 1 shows an example of a functional configuration of an information processing device (computer) according to an embodiment. The information processing device 101 in Fig. 1 includes a storage unit 111 and a calculation unit 112. The storage unit 111 stores model information 121, first evaluation function information 122, and second evaluation function information 123.

[0027] The model information 121 represents a relationship between a plurality of first variables and a plurality of second variables. The first evaluation function information 122 is information for evaluating a first solution including values ​​of the plurality of second variables calculated based on the model information 121. The second evaluation function information 123 is information for evaluating parameters included in the first evaluation function information 122.

[0028] The calculation unit 112 obtains a second solution including values ​​of each of a plurality of second variables from observation information including values ​​of each of a plurality of first variables based on the model information 121, the first evaluation function information 122, and the second evaluation function information 123.

[0029] The first evaluation function information 122 includes first evaluation information indicating the accuracy of the first solution and second evaluation information indicating the sparsity of the first solution. A parameter included in the first evaluation function information 122 indicates the influence of the second evaluation information on the evaluation value of the first solution obtained from the first solution based on the first evaluation function information 122.

[0030] Fig. 2 is a flowchart showing an example of calculation processing performed by the information processing device 101 in Fig. 1. The calculation unit 112 determines parameter values ​​based on the second evaluation function information 123 (step 201), and obtains a second solution based on the determined parameter values ​​(step 202).

[0031] According to the information processing device 101 in FIG. 1, it is possible to obtain a solution having appropriate sparsity for a problem in which a solution is obtained from observed information.

[0032] Fig. 3 shows a specific example of the information processing device 101 in Fig. 1. The information processing device 301 in Fig. 3 includes a preprocessing unit 311, a generating unit 312, a setting unit 313, an estimating unit 314, a selecting unit 315, an output unit 316, and a storage unit 317. The estimating unit 314 and the selecting unit 315 correspond to the calculating unit 112 in Fig. 1, and the storage unit 317 corresponds to the storage unit 111 in Fig. 1.

[0033] The information processing device 301 is used to obtain sparse optimal solutions from observation information in various optimization problems. The optimization problem may be estimation of reflection coefficient distribution in underground radar exploration, basis reduction in regression analysis, or feature selection in neural network.

[0034] The preprocessing unit 311 performs preprocessing such as band limiting and noise removal on the observation information using a band pass filter (window function).

[0035] The generation unit 312 generates an observation model described by equation (2), and stores model information 321 representing the generated observation model in the storage unit 317. The model information 321 includes an observation matrix Φ representing the relationship between vector x and vector y. The vector x is an N-dimensional vector (N is an integer equal to or greater than 1) representing an estimated solution, and the vector y is an M-dimensional vector (M is an integer equal to or greater than 1) representing observation information. The observation matrix Φ is a matrix with M rows and N columns defined from the observation process of the observation information.

[0036] For example, in the case of a problem of estimating an unknown physical quantity from a physical quantity measured by a measurement device, vector x represents the unknown physical quantity, vector y represents the measured physical quantity, and observation matrix Φ is defined from the observation process of the measured physical quantity.

[0037] The model information 321 corresponds to the model information 121 in Fig. 1. The M components of the vector y correspond to a plurality of first variables, and the N components of the vector x correspond to a plurality of second variables.

[0038] Fig. 4 shows an example of observation information in estimating a reflection coefficient distribution. The observation information in Fig. 4 is a three-dimensional array defined by three coordinates: depth, length, and depth. The length and depth represent the measurement range on the earth's surface, and the depth represents the measurement range underground. In this case, M array elements in the depth direction at a given position on the ground determined by a combination of the length coordinate and the depth coordinate are used as vector y representing the physical quantity to be measured.

[0039] In underground radar exploration, a radio wave transmission signal is sent underground from a transmitting antenna of a measuring device installed on the ground, and the radio wave reflected at each position underground is received as a received signal by a receiving antenna. The M array elements in the depth direction respectively represent the received signals reflected at the M positions. For example, the ratio of the strength of the received signal to the strength of the transmitted signal is used as each array element.

[0040] The depth coordinate of each position can be expressed as half the measurement time from when the transmission signal is transmitted to when the reception signal reflected at that position is received. In this case, the depth coordinate of M positions is expressed by, for example, Δt×(j-1) (j=1 to M) using the time interval Δt for receiving the reception signals.

[0041] If the component of vector y at position Δt×(j-1) is y(Δt×(j-1)), then the M components of vector y are y(0), y(Δt), . . . , y(T), where T = Δt×(M-1).

[0042] When estimating the reflection coefficient distribution, N=M, and the reflection coefficient at each of the M positions is used as a vector x that represents an unknown physical quantity. The reflection coefficient represents the ratio of the intensity of the wave reflected from a reference surface to the intensity of the wave incident on the reference surface. Assuming that the ground underground is homogeneous, in theory, a non-zero reflection coefficient can be obtained only on the surface (reflection boundary) of an object such as a buried object.

[0043] The observation information may be a two-dimensional array defined by a depth coordinate and a length or depth coordinate, or may be a one-dimensional array defined by a depth coordinate.

[0044] FIG. 5 shows an example of the distribution of reflection coefficients of the ground where three types of media overlap in the depth direction. Each of medium 501, medium 502, and medium 503 is a homogeneous medium. Waveform 511 represents a received signal, and waveform 512 represents a reflection coefficient. The reflection coefficient is positive, negative, or zero. In this case, the reflection coefficient is zero at positions inside each of medium 501, medium 502, and medium 503, and the reflection coefficient is non-zero only at the boundaries between the media. Therefore, the estimated distribution of reflection coefficients is sparse.

[0045] In actual measurements, the frequency band of radio waves used as transmission signals is finite. For this reason, the waveform 511 of the received signal does not have a pulse shape like the reflection coefficient waveform 512, but has a waveform with a width like a sinc function.

[0046] In this case, the observation matrix Φ is a cyclic convolution matrix with M rows and M columns defined from the transmitted signal, and is expressed by the following equation, for example, using a time interval Δt.

[0047]

number

[0048] The first column of Φ, Φ(0), Φ(Δt), Φ(T), represent the waveform of the transmitted signal. A circulant matrix is ​​a special form of a Toeplitz matrix, in which each row vector is arranged by shifting the elements of the previous row vector by one column. Model information 321 including the observation matrix Φ of equation (4) represents the relationship between M received signals, the waveform of the transmitted signal, and M reflection coefficients.

[0049] Next, the generation unit 312 generates an evaluation function F1 of the formula (3) and an evaluation function F2 that evaluates the regularization parameter λ included in the evaluation function F1, using the model information 321. Then, the generation unit 312 stores, in the storage unit 317, evaluation function information 322 representing the evaluation function F1 and evaluation function information 323 representing the evaluation function F2.

[0050] The evaluation function F1 is a function for evaluating the solution x obtained using the model information 321, and the evaluation function information 322 representing the evaluation function F1 corresponds to the first evaluation function information 122 in Fig. 1. The solution x obtained using the model information 321 corresponds to the first solution, and is expressed as (1 / 2)(L2(Φx-y)) in the formula (3). 2 corresponds to the first evaluation information representing the accuracy of the first solution, and λ(L1(x)) corresponds to the second evaluation information representing the sparsity of the first solution.

[0051] The evaluation value calculated using the evaluation function F1 corresponds to the evaluation value of the first solution. The regularization parameter λ corresponds to a parameter included in the first evaluation function information 122, and represents the influence of λ(L1(x)) on the evaluation value calculated using the evaluation function F1.

[0052] Evaluation function information 323 representing the evaluation function F2 corresponds to the second evaluation function information 123 in Fig. 1. As the evaluation function F2, for example, any one of the evaluation functions shown in the following formulas (5) to (8) is used.

[0053] F2=w1(1-A(λ))+w2(1-B(λ)) (5)

[0054] F2 = w1(A(λ)) + w2(B(λ)) (6)

[0055] F2 = log((1-A(λ)) w1 ×(1-B(λ)) w2 ) (7)

[0056] F2 = (w1 + A(λ)) × (w2 + B(λ)) (8)

[0057] w1 and w2 are positive constants. log() represents the logarithmic function. A(λ) and B(λ) are expressed by the following equations.

[0058] A(λ)=MSE(λ) / max(MSE(λ)) (9) B(λ)=L0(x(λ)) / max(L0(x(λ))) (10)

[0059] MSE(λ) represents the mean squared error (MSE) between y and Φx(λ). max(MSE(λ)) represents the maximum value of MSE(λ) when λ is changed. x(λ) represents the solution x that minimizes the evaluation function F1, and changes depending on the value of λ. L0() represents the L0 norm of a vector. The L0 norm represents the number of non-zero components in a vector. max(L0(x(λ))) represents the maximum value of L0(x(λ)) when λ is changed.

[0060] The larger the MSE(λ), the larger A(λ), and the smaller the MSE(λ), the smaller A(λ). The larger the L0(x(λ)), the larger B(λ), and the smaller the L0(x(λ)), the smaller B(λ).

[0061] In formula (5), w1(1-A(λ)) represents the accuracy of the solution x(λ), and w2(1-B(λ)) represents the sparsity of the solution x(λ). w1(1-A(λ)) is an example of third evaluation information representing the accuracy of the first solution, and w2(1-B(λ)) is an example of fourth evaluation information representing the sparsity of the first solution.

[0062] In formula (6), w1(A(λ)) represents the accuracy of the solution x(λ), and w2(B(λ)) represents the sparsity of the solution x(λ). w1(A(λ)) is an example of third evaluation information representing the accuracy of the first solution, and w2(B(λ)) is an example of fourth evaluation information representing the sparsity of the first solution.

[0063] (1-A(λ)) in equation (7) w1 represents the accuracy of the solution x(λ), and (1-B(λ)) w2 represents the sparsity of the solution x(λ). (1-A(λ)) w1 is an example of the third evaluation information indicating the accuracy of the first solution, and (1-B(λ)) w2 is an example of the fourth evaluation information indicating the sparsity of the first solution.

[0064] In formula (8), w1+A(λ) represents the accuracy of the solution x(λ), and w2+B(λ) represents the sparsity of the solution x(λ). w1+A(λ) is an example of third evaluation information representing the accuracy of the first solution, and w2+B(λ) is an example of fourth evaluation information representing the sparsity of the first solution.

[0065] Instead of the MSE in equation (9), other errors of y and Φx(λ) may be used. The other errors may be, for example, the Root Mean Squared Error (RMSE) or the Mean Absolute Error (MAE). Instead of the L0 norm in equation (10), other indices of sparsity of the solution x(λ) may be used.

[0066] The setting unit 313 sets an upper limit and a lower limit for λ, and sets I discrete values ​​(I is an integer equal to or greater than 1) between the upper limit and the lower limit as candidate values ​​λ(i) (i=1 to I) of λ.

[0067] The estimation unit 314 estimates a solution x that minimizes the evaluation function F1 from a vector y that represents the observation information, using the model information 321 and the evaluation function information 322. The estimation unit 314 estimates a solution x(λ(i)) corresponding to each candidate value λ(i) by using each candidate value λ(i) as λ of the evaluation function F1 and obtaining a solution x(λ(i)) that minimizes the evaluation function F1.

[0068] The estimation unit 314 can estimate the solution x(λ(i)) using an algorithm such as FISTA (Fast Iterative Shrinkage-Thresholding Algorithm) or ADMM (Alternating Direction Method of Multipliers).

[0069] The selection unit 315 obtains an evaluation value for each candidate value λ(i) using the model information 321 and the evaluation function information 323. The selection unit 315 obtains an evaluation value for each candidate value λ(i) by using a solution x(λ(i)) corresponding to each candidate value λ(i) as x(λ) of the evaluation function F2 and calculating the value of the evaluation function F2 from the vector y representing the observation information.

[0070] The maximum value of MSE(λ(1)) through MSE(λ(I)) is used as max(MSE(λ(i))), and the maximum value of L0(x(λ(1))) through L0(x(λ(I))) is used as max(L0(x(λ(i))).

[0071] Next, the selection unit 315 selects the candidate value λ(i) having the optimal evaluation value from among the I candidate values ​​λ(i), and determines the selected candidate value λ(i) as the optimal regularization parameter λopt. Then, the selection unit 315 selects a solution x(λ(i)) corresponding to λopt from among the solutions x(λ(i)) corresponding to each of the I candidate values ​​λ(i), and determines the selected solution x(λ(i)) as the optimal solution xopt. The optimal solution xopt corresponds to the second solution.

[0072] For example, when the evaluation function F2 of the formula (5) or (7) is used, λ(i) that maximizes the evaluation value is determined as λopt. When the evaluation function F2 of the formula (6) or (8) is used, λ(i) that minimizes the evaluation value is determined as λopt. Regardless of which of the evaluation functions F2 of the formulas (5) to (8) is used, it is possible to control λopt by adjusting w1 and w2.

[0073] The output unit 316 outputs the optimal solution xopt as an estimation result of the unknown quantity x. The output unit 316 may output λopt together with the optimal solution xopt. In the case of estimating the reflection coefficient distribution, the j-th (j=1 to M) component of the optimal solution xopt represents an estimate of the reflection coefficient at the position Δt×(j−1).

[0074] According to the information processing device 301 of FIG. 3, an optimal solution having appropriate accuracy and appropriate sparsity can be obtained by selecting the value of the regularization parameter λ included in the evaluation function F1 using the evaluation function F2 that takes into account the accuracy and sparsity of the solution x(λ).

[0075] The information processing device 301 may use an algorithm other than LASSO or an evaluation function F1 other than the formula (3) to find the optimal solution of the formula (2). For example, in the evaluation function F1 of the formula (3), the sum of squares of the components of the vector x may be used instead of the L1 norm of the vector x. (1 / 2)(L2(Φx-y)) in the formula (3) 2 Instead of Φx and y other errors may be used.

[0076] Fig. 6 is a flowchart showing an example of the first optimum solution estimation process performed by the information processing device 301 in Fig. 3. First, the preprocessing unit 311 performs preprocessing such as band limitation and noise removal on the observation information (step 601).

[0077] Next, the generation unit 312 generates an observation model described by equation (2), and stores model information 321 representing the generated observation model in the storage unit 317 (step 602). Then, the generation unit 312 generates evaluation functions F1 and F2 using the model information 321, and stores evaluation function information 322 representing the evaluation function F1 and evaluation function information 323 representing the evaluation function F2 in the storage unit 317 (step 603).

[0078] Next, the setting unit 313 sets I candidate values ​​λ(i) (i=1 to I) of λ (step 604).

[0079] Next, the estimation unit 314 sets the control variable i to 1 (step 605), and estimates a solution x(λ(i)) that minimizes the evaluation function F1 by using the candidate value λ(i) as the λ of the evaluation function F1 (step 606). Then, the estimation unit 314 compares i with I (step 607).

[0080] If i is smaller than I (step 607, NO), the estimation unit 314 increments i by 1 (step 608) and repeats the processes from step 606 onwards.

[0081] If i=I (step 607, YES), the selection unit 315 uses the solutions x(λ(i)) corresponding to each of the candidate values ​​λ(1) to λ(I) as x(λ) of the evaluation function F2 to calculate the value of the evaluation function F2 (step 609).

[0082] Next, the selection unit 315 selects the candidate value λ(i) having the optimal evaluation value from among the I candidate values ​​λ(i), and determines it as the optimal regularization parameter λopt (step 610). Next, the selection unit 315 selects a solution x(λ(i)) corresponding to λopt from among the solutions x(λ(i)) corresponding to each of the I candidate values ​​λ(i), and determines it as the optimal solution xopt (step 611). Then, the output unit 316 outputs the optimal solution xopt (step 612).

[0083] Figures 7A and 7B are flowcharts showing an example of a second optimal solution estimation process performed by the information processing device 301 in Figure 3. In the optimal solution estimation process in Figures 7A and 7B, an optimal solution xopt is estimated using K-fold cross validation.

[0084] The processing in steps 701 to 704 is similar to the processing in steps 601 to 604 in FIG.

[0085] After I candidate values ​​λ(i) are set, the estimation unit 314 divides the vector y representing the observation information into K vectors (K is an integer equal to or greater than 2) to generate K vectors v(k) (k=1 to K) (step 705). Each vector v(k) is made up of a portion of the components of the vector y.

[0086] Next, the estimation unit 314 sets the control variable k to 1 (step 706), selects the vector v(k) as the validation data p(k), and selects K-1 vectors v(k) other than the validation data as the training data q(k) (step 707). Therefore, the training data q(k) is a vector consisting of the remaining components of the M components of the vector y excluding the components included in the validation data p(k).

[0087] Next, the estimation unit 314 sets the control variable i to 1 (step 708), and estimates a solution x(λ(i), k) that minimizes the evaluation function F1 by using the candidate value λ(i) as λ of the evaluation function F1 (step 709). In step 709, the training data q(k) is used instead of the vector y in equation (3), and the observation matrix φ(k) is used instead of the observation matrix Φ. The observation matrix φ(k) is a matrix consisting of only the rows corresponding to each component of the training data q(k) among the M rows of the observation matrix Φ.

[0088] Next, the estimation unit 314 calculates the mean square error MSE(λ(i),k) between the validation data p(k) and τ(k)x(λ(i),k) (step 710). τ(k) is a matrix consisting of only the rows corresponding to each component of the validation data p(k) among the M rows of the observation matrix Φ. Then, the estimation unit 314 compares i with I (step 711).

[0089] If i is smaller than I (step 711, NO), the estimation unit 314 increments i by 1 (step 712) and repeats the processes from step 709 onwards.

[0090] If i=I (step 711, YES), the estimation unit 314 compares k with K (step 713). If k is smaller than K (step 713, NO), the estimation unit 314 increments k by 1 (step 714) and repeats the processes from step 707 onwards.

[0091] If k=K (step 713, YES), the estimation unit 314 calculates the average of x(λ(i),1) to x(λ(i),K) and determines the calculated average as the solution x(λ(i)) (step 715).Then, the estimation unit 314 calculates the average of MSE(λ(i),1) to MSE(λ(i),K) and determines the calculated average as MSE(λ(i)) (step 716).

[0092] The processing in steps 717 to 720 is similar to the processing in steps 609 to 612 in FIG.

[0093] In the optimal solution estimation process of FIGS. 7A and 7B, other cross-validation methods may be used instead of the K-fold cross-validation.

[0094] Figure 8 shows an example of a first estimation result of the reflection coefficient distribution in a ground-penetrating radar survey. In this example, a steel pipe is buried underground, and the surface of the steel pipe is detected as a reflection boundary, along with the boundary between media.

[0095] FIG. 8(a) shows the result of estimating the reflection coefficient distribution using the evaluation function F2 that includes only the first term of equation (5). The horizontal axis represents the length of FIG. 4, and the vertical axis represents the depth of FIG. 4. The sign "+" indicates the position of the estimated non-zero reflection coefficient. FIG. 8(b) shows the result of estimating the reflection coefficient distribution using the evaluation function F2 that includes both the first and second terms of equation (5).

[0096] In the estimation result in Fig. 8(a), the reflection coefficients are distributed all over the area, but in the estimation result in Fig. 8(b), the number of reflection coefficients is reduced and the reflection coefficients are concentrated near the ground surface and on the surface of the steel pipe. Therefore, it can be seen that the surface of the steel pipe can be more clearly confirmed by using the evaluation function F2, which adds the second term of equation (5).

[0097] Figure 9 shows an example of the second estimation result of the reflection coefficient distribution in a ground-penetrating radar survey. In this example, a cavity exists underground, and the surface of the cavity is detected as a reflection boundary along with the boundary between media.

[0098] FIG. 9(a) shows the result of estimating the reflection coefficient distribution using the evaluation function F2 that includes only the first term of equation (5). The horizontal axis represents the length of FIG. 4, and the vertical axis represents the depth of FIG. 4. The sign "+" indicates the position of the estimated non-zero reflection coefficient. FIG. 9(b) shows the result of estimating the reflection coefficient distribution using the evaluation function F2 that includes both the first and second terms of equation (5).

[0099] In the estimation result in Fig. 9(a), the reflection coefficients are distributed all over the area, but in the estimation result in Fig. 9(b), the number of reflection coefficients is reduced and the reflection coefficients are concentrated near the ground surface and on the surface of the cavity. Therefore, it can be seen that the surface of the cavity can be more clearly confirmed by using the evaluation function F2, which adds the second term of equation (5).

[0100] The information processing device 301 in FIG. 3 is capable of not only estimating the reflection coefficient distribution but also finding optimal solutions to other optimization problems such as basis reduction in regression analysis and feature selection in neural networks.

[0101] In the case of regression analysis, the observed data of the objective variable is used as vector y, and the regression coefficients of each of the multiple explanatory variables are used as vector x. In addition, multiple basis functions such as polynomial basis and Fourier basis used as the explanatory variables are used as the observation matrix Φ. For example, in the case of polynomial regression, the observation matrix Φ may be a Vandermonde matrix of the basis functions. By estimating a sparse solution x, the number of bases included in the regression equation can be reduced.

[0102] In the case of image reconstruction using a convolutional neural network, the input image is used as vector y, and the weights for each of the multiple features (filters) in the convolutional layer are used as vector x. In addition, the multiple features in the convolutional layer are used as the observation matrix Φ. Image reconstruction is used, for example, to detect anomalies from an input image.

[0103] The configurations of the information processing device 101 in Fig. 1 and the information processing device 301 in Fig. 3 are merely examples, and some of the components may be omitted or changed depending on the application or conditions of the information processing device. For example, in the information processing device 301 in Fig. 3, when it is not necessary to perform band limiting, noise removal, etc. on the observation information, the pre-processing unit 311 can be omitted.

[0104] The flowcharts shown in Figures 2, 6, 7A, and 7B are merely examples, and some of the processes may be omitted or changed depending on the configuration or conditions of the information processing device. For example, in the optimal solution estimation process of Figure 6 and the optimal solution estimation process of Figures 7A and 7B, if there is no need to perform band limiting, noise removal, etc. on the observation information, the pre-processing of steps 601 and 701 can be omitted.

[0105] The observation information shown in Fig. 4 is merely an example, and the observation information changes depending on the use or conditions of the information processing device. The ground reflection coefficient distribution shown in Fig. 5 is merely an example, and the reflection coefficient distribution changes depending on the underground structure, the presence or absence of buried objects, etc. The estimation results of the reflection coefficient distribution shown in Fig. 8 and Fig. 9 are merely an example, and the estimation results of the reflection coefficient distribution change depending on the underground structure, the shape of buried objects, etc.

[0106] Equations (1) to (10) are merely examples, and the information processing device 301 may perform the optimal solution estimation process using other calculation equations.

[0107] Fig. 10 shows an example of the hardware configuration of the information processing device 101 in Fig. 1 and the information processing device 301 in Fig. 3. The information processing device in Fig. 10 includes a CPU (Central Processing Unit) 1001, a memory 1002, an input device 1003, an output device 1004, an auxiliary storage device 1005, a media drive device 1006, and a network connection device 1007. These components are hardware and are connected to each other via a bus 1008.

[0108] The memory 1002 is, for example, a semiconductor memory such as a Read Only Memory (ROM) or a Random Access Memory (RAM), and stores programs and data used in processing. The memory 1002 may operate as the storage unit 111 in FIG. 1 or the storage unit 317 in FIG.

[0109] 1 by executing a program using the memory 1002. The CPU 1001 (processor) also operates as the preprocessing unit 311, the generating unit 312, the setting unit 313, the estimating unit 314, and the selecting unit 315 in FIG.

[0110] The input device 1003 is, for example, a keyboard, a pointing device, etc., and is used to input instructions or information from a user or operator. The output device 1004 is, for example, a display device, a printer, a speaker, etc., and is used to output inquiries or instructions to a user or operator, and processing results. The output device 1004 may operate as the output unit 316 in Fig. 3. The processing result may be the optimal solution xopt, or may be the optimal solutions xopt and λopt.

[0111] The auxiliary storage device 1005 is, for example, a magnetic disk device, an optical disk device, a magneto-optical disk device, a tape device, or the like. The auxiliary storage device 1005 may be a hard disk drive or an SSD (Solid State Drive). The information processing device can store programs and data in the auxiliary storage device 1005 and use them by loading them into the memory 1002. The auxiliary storage device 1005 may operate as the storage unit 111 in FIG. 1 or the storage unit 317 in FIG. 3.

[0112] The medium drive device 1006 drives the portable recording medium 1009 and accesses the recorded contents. The portable recording medium 1009 is a memory device, a flexible disk, an optical disk, a magneto-optical disk, etc. The portable recording medium 1009 may be a CD-ROM (Compact Disk Read Only Memory), a DVD (Digital Versatile Disk), a USB (Universal Serial Bus) memory, etc. A user or operator can store programs and data in the portable recording medium 1009 and load them into the memory 1002 for use.

[0113] In this manner, the computer-readable recording medium that stores the program and data used in the processing is a physical (non-transitory) recording medium such as the memory 1002, the auxiliary storage device 1005, or the portable recording medium 1009.

[0114] The network connection device 1007 is a communication interface circuit that is connected to a communication network such as a WAN (Wide Area Network) or a LAN (Local Area Network) and performs data conversion associated with communication. The information processing device can receive programs and data from an external device via the network connection device 1007 and load them into the memory 1002 for use. The network connection device 1007 may operate as the output unit 316 in FIG. 3.

[0115] It is not necessary for the information processing device to include all of the components in Fig. 10, and some of the components may be omitted depending on the application or conditions. For example, if an interface with a user or operator is not required, the input device 1003 and the output device 1004 may be omitted. If the portable recording medium 1009 or a communication network is not used, the medium drive device 1006 or the network connection device 1007 may be omitted.

[0116] Although the disclosed embodiments and their advantages have been described in detail, it will be appreciated that those skilled in the art may make various modifications, additions and omissions therein without departing from the scope of the invention as clearly set forth in the claims. [Explanation of symbols]

[0117] 101, 301 Information processing device 111, 317 Memory section 112 Calculation Department 121, 321 Model Information 122 First evaluation function information 123 Second Evaluation Function Information 311 Pretreatment section 312 Generation part 313 Settings 314 Estimation Department 315 Selection Section 316 Output section 322, 323 Evaluation function information 501~503 Medium 511, 512 waveform 1001 CPU 1002 Memory 1003 Input Device 1004 Output device 1005 Auxiliary storage device 1006 Media drive unit 1007 Network connection device 1008 Bus 1009 Portable recording media

Claims

1. a storage unit that stores model information that indicates a relationship between a plurality of first variables and a plurality of second variables, first evaluation function information that evaluates a first solution including values ​​of each of the plurality of second variables calculated based on the model information, and second evaluation function information that evaluates a parameter included in the first evaluation function information; a calculation unit that determines a second solution including values ​​of each of the plurality of second variables from observation information including values ​​of each of the plurality of first variables based on the model information, the first evaluation function information, and the second evaluation function information; Equipped with the first evaluation function information includes first evaluation information representing accuracy of the first solution and second evaluation information representing sparsity of the first solution; the parameter represents a degree of influence of the second evaluation information on an evaluation value of the first solution obtained from the first solution based on the first evaluation function information; The information processing apparatus according to claim 1, wherein the calculation unit determines a value of the parameter based on the second evaluation function information, and obtains the second solution based on the determined value of the parameter.

2. 2. The information processing device according to claim 1, wherein the second evaluation function information includes third evaluation information representing an accuracy of the first solution that changes depending on the parameter, and fourth evaluation information representing a sparsity of the first solution that changes depending on the parameter.

3. 3. The information processing device according to claim 2, wherein the calculation unit determines the first solution from the observation information based on the first evaluation function information including each of a plurality of candidate values ​​of the parameter, determines an evaluation value for each of the plurality of candidate values ​​from the first solution based on the second evaluation function information, determines one of the plurality of candidate values ​​as the value of the parameter based on the evaluation value for each of the plurality of candidate values, and determines the first solution determined based on the first evaluation function information including the determined value of the parameter as the second solution.

4. The plurality of first variables represent received signals that are received as a result of a transmission signal being reflected; the plurality of second variables represent a plurality of reflection coefficients; 4. The information processing apparatus according to claim 1, wherein the model information represents a relationship between the waveforms of the received signal and the transmitted signal and the plurality of reflection coefficients.

5. a computer executes a process of obtaining a second solution including values ​​of each of the second variables from observation information including values ​​of each of the first variables, based on model information indicating a relationship between a plurality of first variables and a plurality of second variables, first evaluation function information for evaluating a first solution including values ​​of each of the second variables obtained based on the model information, and second evaluation function information for evaluating a parameter included in the first evaluation function information; the first evaluation function information includes first evaluation information representing accuracy of the first solution and second evaluation information representing sparsity of the first solution; the parameter represents a degree of influence of the second evaluation information on an evaluation value of the first solution obtained from the first solution based on the first evaluation function information; The process of obtaining the second solution includes: A process of determining a value of the parameter based on the second evaluation function information; determining the second solution based on the determined parameter values; 13. An information processing method comprising:

6. a process of determining a second solution including values ​​of each of the second variables from observation information including values ​​of each of the first variables, based on model information indicating a relationship between a plurality of first variables and a plurality of second variables, first evaluation function information for evaluating a first solution including values ​​of each of the second variables determined based on the model information, and second evaluation function information for evaluating a parameter included in the first evaluation function information; the first evaluation function information includes first evaluation information representing accuracy of the first solution and second evaluation information representing sparsity of the first solution; the parameter represents a degree of influence of the second evaluation information on an evaluation value of the first solution obtained from the first solution based on the first evaluation function information; The process of obtaining the second solution includes: A process of determining a value of the parameter based on the second evaluation function information; determining the second solution based on the determined parameter values; A program comprising:

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