Related factor extraction device, related factor extraction method and program

Abstract

To provide a related factor extraction apparatus, method, and program for extracting a small number of factors having correlations between dimensions from multidimensional tensor data with defects.SOLUTION: A related factor extraction apparatus 1 comprises: a defected element designation unit 21 for designating a defected element; a factorization calculation unit 22 for obtaining two-dimensional tensor data by multiplying three-dimensional tensor data by an orthonormal matrix and performing singular value decomposition of the two-dimensional tensor data to factorize the three-dimensional tensor data; a reduction direction calculation unit 23 for calculating a new orthonormal matrix that reduces the sum of factorization coefficients for the orthonormal matrix; a provisional restoration unit 24 for provisionally restoring multidimensional tensor data; and a consistency checking unit 25 for matching the restored multidimensional tensor data and observed value data. The apparatus repeats, using the restored data, a series of calculations by the factorization calculation unit, the reduction direction calculation unit, the provisional restoration unit, and the consistency checking unit until a predetermined termination condition is satisfied in order to obtain relevant factors of the three-dimensional tensor data.SELECTED DRAWING: Figure 1

Description

[Technical Field]

[0001] The present invention relates to an apparatus for extracting factors having correlations between dimensions from multidimensional data with missing data. [Background technology]

[0002] Real-world information and measurement data obtained using analytical equipment can have a multidimensional data structure. Multidimensional data is called a "tensor." For example, a time-series signal has spatial and temporal dimensions, and a spectroscopic image has spatial and wavelength dimensions. In fluorescence analysis, if the incident light wavelength and fluorescence wavelength are measured at the mixture ratio of various components, they can be expressed as a three-dimensional tensor. In addition, there can be correlations between various labels that characterize products and their target demographics, and these can also be expressed as tensors.

[0003] The structure of a multidimensional tensor becomes more complex as the number of dimensions increases, making it more difficult to extract essential information from it. The essence of the information sometimes appears as a factor that is correlated between dimensions. Here, we will explain the quantification of the mixing ratio of fluorescent components in a medium in fluorescence analysis as an example. In this example, a mixture is prepared in which multiple fluorescent components are mixed at a certain ratio. Several mixtures are prepared in which the components remain the same but the mixing ratio is changed. For example, suppose there is a mixture in which the mixing ratio of fluorescent components A, B, and C is changed in 10 different ways.

[0004] When this mixture is analyzed for fluorescence, three-dimensional tensor data is obtained by recording the wavelengths of incident light and fluorescence. One dimension is a number that specifies the mixture, which takes on a natural value between 1 and 10. The remaining two dimensions are the wavelengths of incident light and fluorescence, respectively, and take on real values ​​representing the wavelengths used in the measurement. For example, suppose 100 values ​​are recorded for incident light and fluorescence. The three-dimensional tensor index (1, 11, 13) represents the intensity at the 11th incident light wavelength (say 400 nm) for the first mixture, and the intensity at the 13th recorded wavelength (say 600 nm) of the fluorescence wavelengths observed at that time. Here, the first index takes on 10 values, but because all mixtures only have the common fluorescence components A, B, and C, the first dimension of the three-dimensional tensor data, which is 10 x 100 x 100 in size, is expected to be represented by a smaller number of factors. Furthermore, for the remaining two dimensions, the wavelengths of incident light and fluorescence are in a physicochemical causal relationship with each other, and each fluorescence component has its own unique correlation, so it is expected that the remaining two dimensions can be expressed with fewer factors than 100 x 100.

[0005] Therefore, if the tensor data of fluorescence analysis is expressed as the sum of these correlated factors, the weight ratio of the sum will be the mixture rate of the fluorescent components. Generalizing this, in a multidimensional data structure, the small number of factors that are correlated between dimensions are the essence of the information, and it is considered important to extract these factors to express the data structure. If a tensor data structure can be expressed in this way, it is believed that the essence of the information can be extracted. Incidentally, Patent Document 1 discloses a multidimensional correlation data extraction device, but does not take into account missing data. [Prior art documents] [Patent documents]

[0006] [Patent Document 1] Patent Publication No. 2017-10438 Summary of the Invention [Problem to be solved by the invention]

[0007] Real data often contains missing data, and in such cases, it is necessary to extract factors from multidimensional data with missing data. For example, in the fluorescence analysis mentioned above, it takes a long time to observe 100 x 100 different wavelengths when observing the wavelengths of incident light and fluorescence, so suppose that only a few different wavelengths are observed for each mixture. In this case, the fluorescence intensity for incident light at wavelengths that are not observed becomes a missing value. In this case, it is necessary to somehow complement the missing values, and for the complemented data to be represented as the sum of a small number of factors that are correlated between dimensions.

[0008] In view of the above background, the present invention aims to provide a technique for extracting a small number of factors that have correlations between dimensions from multidimensional tensor data with missing data. The difficulty in this problem is that the original structure of the missing data is unknown, and the method for decomposing the data into the desired factors is uncertain. [Means for solving the problem]

[0009] The related factor extraction device of the present invention is a device for extracting related factors from observation data of a missing multidimensional tensor, and includes: a missing element designation unit that designates missing elements of the multidimensional tensor data; a factorization calculation unit that multiplies (number of dimensions - 2) dimensions of the multidimensional tensor data by an orthonormal matrix to generate two-dimensional tensor data and performs singular value decomposition on the two-dimensional tensor data to factorize the multidimensional tensor data; a reduction direction calculation unit that calculates a gradient direction that reduces the sum of the factorization coefficients for the orthonormal matrix and calculates a new orthonormal matrix in that gradient direction; and a processing unit that performs processing on the factorization calculation unit and the reduction calculation unit. The present invention includes a tentative restoration unit that restores multidimensional tensor data using the orthonormal matrix and the result of singular value decomposition calculated by the direction calculation unit, and a consistency checking unit that performs processing to align the multidimensional tensor data restored by the tentative restoration unit with observed value data, and a series of calculations by the factorization calculation unit, the reduction direction calculation unit, the tentative restoration unit, and the consistency checking unit are repeated using the restored data that has been subjected to the alignment processing by the consistency checking unit until a predetermined termination condition is satisfied, and when the predetermined termination condition is satisfied, the result of the factorization by the factorization calculation unit is obtained as a related factor of the multidimensional tensor data.

[0010] Although multidimensional tensor data cannot be uniquely factorized, the configuration of the present invention decomposes the multidimensional tensor data using an orthonormal matrix. By updating the orthonormal matrix to reduce the sum of the factorization coefficients and ensuring consistency with the observed value data, factorization can be performed while complementing missing data, and related factors of the multidimensional data can be found.

[0011] In the associated factor extraction device of the present invention, the reduction direction calculation unit may add a vector indicating the gradient direction to a current orthonormal matrix, and map the added matrix into a space in which orthonormal matrices exist, to obtain a new orthonormal matrix.

[0012] By mapping to a space where orthonormal matrices exist, it is possible to obtain an orthonormal matrix with a smaller sum of the factorization coefficients. Note that the mapping method can be an exponential mapping or retraction.

[0013] In the relevant factor extraction device of the present invention, the reduction direction calculation unit may use a backtracking method to find a gradient direction in which the sum of the factorization coefficients is equal to or smaller than a predetermined threshold.

[0014] In the associated factor extraction device of the present invention, the consistency checking unit may replace values ​​of the restored multidimensional tensor data with the observed value data for tensor elements for which the observed value data exists.

[0015] In the associated factor extraction device of the present invention, the consistency checking unit may replace a tensor element for which the observed value data exists with the observed value when an error between the value of the restored multidimensional tensor data and the observed value is equal to or greater than a threshold.

[0016] In the related factor extraction device of the present invention, when an error between a value of the restored multidimensional tensor data and the observed value data for a tensor element for which the observed value data exists is equal to or greater than a threshold, the consistency checking unit may replace the tensor element with a value determined based on the observed value and the value of the multidimensional tensor data.

[0017] The related factor extraction method of the present invention is a method for extracting related factors from observation data of a missing multidimensional tensor, and includes a first step of specifying missing elements of the multidimensional tensor data, a second step of multiplying (the number of dimensions - 2) dimensions of the multidimensional tensor data by an orthonormal matrix to obtain two-dimensional tensor data, and factorizing the two-dimensional tensor data by singular value decomposition, and a third step of calculating a gradient direction that reduces the sum of the factorization coefficients for the orthonormal matrix and calculating a new orthonormal matrix in the gradient direction, and a third step of performing the second step and the third step. a fourth step of reconstructing the multidimensional tensor data using the orthogonal matrix and the result of singular value decomposition calculated in the previous step, and a fifth step of performing processing to align the multidimensional tensor data reconstructed in the fourth step with observed value data, wherein the second step, the third step, the fourth step, and the fifth step are repeated using the reconstructed data that has been subjected to processing to align it with the observed value data in the fifth step until a predetermined termination condition is met, and when the predetermined termination condition is met, the result of the factorization calculated in the second step is calculated as the associated factor of the multidimensional tensor data.

[0018] The program of the present invention is a program for extracting related factors from observation data of a missing multidimensional tensor, and includes a first step of specifying missing elements of the multidimensional tensor data, a second step of multiplying the (number of dimensions - 2) dimensions of the multidimensional tensor data by an orthonormal matrix to obtain two-dimensional tensor data, and factorizing the two-dimensional tensor data by singular value decomposition, a third step of calculating a gradient direction that reduces the sum of the factorization coefficients for the orthonormal matrix, and calculating a new orthonormal matrix in that gradient direction, and a third step of performing the second step and the second step. A fourth step is executed in which multidimensional tensor data is reconstructed using the orthonormal matrix and the result of singular value decomposition calculated in the third step, and a fifth step is executed in which the multidimensional tensor data reconstructed in the fourth step is aligned with observed value data. The second step, the third step, the fourth step, and the fifth step are repeated using the reconstructed data that has been aligned with the observed value data in the fifth step until a predetermined termination condition is met, and when the predetermined termination condition is met, the result of the factorization obtained in the second step is obtained as the associated factor of the multidimensional tensor data. [Effects of the Invention]

[0019] According to the present invention, a small number of factors having correlations between dimensions can be extracted from multidimensional tensor data with missing data. [Brief explanation of the drawings]

[0020] [Figure 1] FIG. 1 illustrates a configuration of a related factor extraction device according to an embodiment. [Figure 2] FIG. 10 is a diagram for conceptually explaining calculations performed by a factorization calculation unit. [Figure 3] FIG. 10 is a diagram illustrating an image of a method for updating and calculating an orthonormal matrix. [Figure 4] FIG. 10 is a diagram illustrating the operation of the related factor extraction device according to the embodiment. DETAILED DESCRIPTION OF THE INVENTION

[0021] A related factor extraction device 1 according to an embodiment of the present invention will be described below with reference to the drawings. In the following embodiment, an apparatus that extracts related factors from three-dimensional fluorescence analysis data will be described as an example, but as will be described later, it can be extended to a multidimensional tensor structure.

[0022] FIG. 1 is a diagram showing the configuration of a related factor extraction device 1 according to an embodiment. The related factor extraction device 1 includes a data acquisition unit 10 that acquires three-dimensional tensor data to be analyzed, a calculation unit 11 that extracts related factors from the input three-dimensional tensor data, a memory unit 12 that stores the calculation results, and an output unit 13 that outputs the calculation results. The calculation unit 11 includes a missing element designation unit 21, a factorization calculation unit 22, a reduction direction calculation unit 23, a tentative restoration unit 24, and a consistency check unit 25. Each component will be described below.

[0023] The missing element designation unit 21 has the function of designating missing elements of input tensor data. In the following explanation, elements of tensor data X are expressed by natural number indexes. For example, if an element of a three-dimensional tensor is written as (i1, i2, i3), X(i1, i2, i3) represents the value stored in that element. The size of each dimension is M j For example, in the jth dimension (j=1,2,3), j =1, ,M j The missing element designation unit 21 defines the index set of the observed data of the tensor as Ω={(i1, i2, i3)|i j ∈N, j=1, 2, 3}, and the value of the tensor at the index of the complement set is set to 0. That is, the missing element designation unit 21 performs processing to set the value of missing elements with no observed data to 0.

[0024] The factorization calculation unit 22 has the function of factorizing three-dimensional tensors. Unlike various matrix factorizations, tensor factorization methods are generally not unique. In this device, tensors are factorized according to the orthonormal tensor decomposition method described in the following reference:

[0025] (References) A. Franc, ``Etude alge'brique des multitableaux : apports de l'alge`bre tensorielle'' These de Doctorat, 1992. TG Kolda, “Orthogonal tensordecompositions,” SIAM Journal on Matrix Analysis and Applications, vol. 23, no. 1, 2003.

[0026] When the number of factors is K, the orthonormal tensor decomposition of the tensor X is given by

number

[0027] where x3 (k) is a vector of size M3, which represents the information on the mixing ratio of the fluorescent components. For one k, x j (s,k) (i j ) is the size M j (where j=1,2), and the parentheses indicate the indices. s,k are summed with the rank of the tensor obtained by this factorization as the upper limit. The dot product of two vectors is orthonormal x j (k) x j (l) =δ kl where δ kl is 1 if k=l and 0 if k≠1.

[0028] The remaining two-dimensional vectors represent the information on the observed wavelengths of the incident light and the fluorescence, respectively.

number

[0029] is one factor. Generally, such factorization cannot be uniquely calculated for tensors with three or more dimensions. Therefore, the factorization calculation unit 22 converts the three-dimensional tensor into a two-dimensional tensor and then performs singular value decomposition of the two-dimensional tensor. Specifically, the factorization calculation unit 22 receives the tensor X and an orthonormal matrix R3 of the size of the third dimension at the same time, and the factorization calculation unit 22 performs the following calculation.

[0030] The transpose of the orthonormal matrix R3 (which is also the inverse) is applied to the third dimension of the tensor X (x3 is multiplied by R3 from the left), resulting in X ~ (X tilde: The symbol is "~" above X, but for the sake of the specification, it is "X ~ ") X ~ The 2D tensor with the third dimension index fixed, that is, the matrix spanned by the first and second dimensions, is subjected to singular value decomposition. k is positive and can be calculated uniquely.

[0031] Here, the purpose of transforming with the orthonormal matrix R3 is to convert a mixture with unknown mixing ratios into the simplest possible expression. For example, it is to transform it into a sum of factors that represent the causal relationship between incident light and fluorescence for the mixed fluorescent components A, B, and C themselves or their representative mixed states. In this way, the factorization calculation unit 22 uniquely outputs the result of orthonormal tensor decomposition from the provided tensor X and one orthonormal matrix R3. Here, missing values ​​of tensor X are provisionally imputed simultaneously with the factorization.

[0032] 2(a) and 2(b) are diagrams for conceptually explaining the calculations performed by the factorization calculation unit 22. Fig. 2(a) is a schematic diagram showing fluorescence analysis data as an example of three-dimensional tensor data X. It shows that intensity data corresponds to coordinates indicated by the indexes of the mixture number, incident light wavelength, and fluorescence wavelength.

[0033] Figure 2(b) shows a schematic diagram of the transformation of a mixture of tensor data by multiplying the axis of the mixture by the transpose of the orthonormal matrix R3. This results in a projection into a latent low-dimensional vector space basis that represents the mixture. Since this space basis is two-dimensional, singular value decomposition can be performed, and by performing singular value decomposition, we can obtain the tensor decomposition when a certain orthonormal matrix R3 is applied.

[0034] The reduction direction calculation unit 23 calculates an orthonormal matrix R3 that reduces the number of factors in order to express the tensor with a smaller number of factors. Specifically, the reduction direction calculation unit 23 calculates an orthonormal matrix R3 that reduces the number of factors. k sum over k of

number

[0035] First, the reduction direction calculation unit 23 calculates S for each element of R3 in the matrix space expanded to the Euclidean space. σ Calculate the derivative of . Let this derivative be vector A.

number

[0036] The matrix is ​​changed according to this gradient A and projected onto an orthogonal matrix space. Exponential mapping and retraction can be used to calculate the projection. Retraction is described in the following references.

[0037] (References) PA Absil, R. Mahony, and R. Sepulchre, “Optimization algorithms on matrix manifolds” Princeton University Press, 2008.

[0038] FIG. 3 is a diagram showing an image of a method for updating and calculating the orthonormal matrix R3. Current value

number

number

number

[0039] S σ The method of updating is determined when the value of becomes sufficiently small. This can be done using backtracking, etc. Backtracking is a method of searching for a solution by trial and error using combinations of variable values.

[0040] The number of factors having non-zero coefficients in the orthonormal tensor decomposition using the orthonormal matrix updated by the reduction direction calculation unit 23, or the number of factors having sufficiently large coefficients, is expected to be smaller than before the update by the reduction direction calculation unit 23. s,k is a positive value, and the sum of them is S σ Reducing the value of is approximately equivalent to reducing the number of factors.

[0041] The provisional restoration unit 24 has a function of provisionally restoring all elements of a tensor based on the tensor structure output via the factorization calculation unit 22 and the reduction direction calculation unit 23. The tensor structure has a simplicity that depends on the factorization.

[0042] The consistency checking unit 25 has a function of checking the consistency between the tensor elements restored by the tentative restoration unit 24 and the observed values, and correcting any discrepancies. Since the tensor elements restored by the tentative restoration unit 24 are not necessarily consistent with the observed values, the consistency checking unit 25 updates Ω, which is the set of observed indexes among the tensor elements, so that the observed values ​​and the element values ​​of the tensor output by the consistency checking unit 25 are consistent. That is, for the indexes of Ω, the tensor values ​​output by the factorization calculation unit 22 are updated so that they have a high affinity with the observed values. In this embodiment, the consistency checking unit 25 replaces the element values ​​of the tensor at the index of Ω at the time of the current update with the observed values. This is a method of assuming that the observed values ​​are noise-free and the observed value data are true values.

[0043] The memory unit 12 has a function of storing data with missing parts complemented and orthonormal tensor decomposition results that express the data using decimal points. The calculation unit 11 stores the complemented data and the orthonormal tensor decomposition results in the memory unit 12 when the differences between the inputs and outputs (orthonormal matrix R and tensors) of the factorization calculation unit 22, reduction direction calculation unit 23, and provisional reconstruction unit 24 become sufficiently small. Here, the differences between the inputs and outputs of the factorization calculation unit 22, reduction direction calculation unit 23, and provisional reconstruction unit 24 are used as termination conditions. However, this is because, for example, even if the difference between the inputs and outputs of the orthonormal matrix R becomes small, there may still be differences between the inputs and outputs of the factorization calculation unit 22 and provisional reconstruction unit 24, or vice versa. However, it is not essential to use the differences between the inputs and outputs of the factorization calculation unit 22, reduction direction calculation unit 23, and provisional reconstruction unit 24 as termination conditions; the differences between any one or two of the inputs and outputs may also be used.

[0044] The configuration of the related factor device of this embodiment has been described above, but an example of the hardware of the related factor extraction device 1 described above is a computer equipped with a CPU, RAM, ROM, hard disk, display, keyboard, mouse, communication interface, etc. A program having modules that realize each of the above functions is stored in RAM or ROM, and the related factor extraction device 1 is realized by executing the program with a CPU. Such programs are also included in the scope of the present invention.

[0045] 4 is a flowchart showing the operation of the related factor extraction device 1 according to the embodiment. The related factor extraction device 1 acquires three-dimensional tensor data to be analyzed (S10). In this embodiment, three-dimensional fluorescence analysis data is acquired. The related factor extraction device 1 inputs 0 to missing tensor elements in the missing element acquisition unit. Missing tensor elements are typically unobserved tensor elements.

[0046] Next, the related factor extraction device 1 performs factorization calculations on the tensor data (S12). Because the factorization of three-dimensional tensor data is not unique, as described above, the three-dimensional tensor is converted into a two-dimensional tensor by applying the orthonormal matrix R3, and then the two-dimensional tensor is subjected to singular value decomposition to perform factorization.

[0047] Next, the related factor extraction device 1 calculates the direction of reduction of the number of factors (S13). That is, the orthonormal matrix R3 is updated so that the sum of the factorization coefficients becomes smaller. The related factor extraction device 1 tentatively restores all elements of the tensor based on the tensor structure output through the factorization calculation and the reduction direction calculation (S14).

[0048] The associated factor extraction device 1 checks the consistency between the provisionally restored tensor elements and the observed values ​​(S15). Specifically, for tensor elements with observed values, the associated factor extraction device 1 replaces the values ​​of the provisionally restored tensor elements with the observed values.

[0049] The associated factor extraction device 1 determines whether to end the process (S16). The associated factor extraction device 1 determines whether the input / output differences in the factorization calculation unit 22, reduction direction calculation unit 23, and provisional restoration unit 24 are smaller than a predetermined threshold, and ends the process when they are smaller than the threshold (YES in S16), and stores the provisional restoration result (i.e., complementary data) and the factorization result in the storage unit 12.

[0050] If the process is not to be ended (NO in S16), the associated factor extraction device 1 returns to the process of performing factorization calculation (S12). The associated factor extraction device 1 repeats the process from factorization calculation (S12) to consistency check of observed values ​​(S15) using the provisionally restored tensor data.

[0051] The related factor extraction device 1 of this embodiment can perform factorization to represent data with a small number of elements while complementing missing tensor data, thereby making it possible to grasp the data structure of a multidimensional tensor even from partial observations.

[0052] Although the related factor extraction device of the present invention has been described in detail above by way of an embodiment, the related factor extraction device of the present invention is not limited to the above-described embodiment.

[0053] In the above embodiment, the consistency checking unit 25 corrects data by prioritizing the observed value over a provisionally restored value for a tensor element with an observed value. However, noise in the observed value may also be taken into consideration. Specifically, a range is set around the observed value, and the value of the observed element is updated within that range. This method is based on the idea that the latent simple structure of the data extracted by the factorization calculation unit 22 and reduction direction calculation unit 23 has high priority, and the error range of the observed value is acceptable as long as the simple structure can be described.

[0054] As an example, the consistency checking unit 25 compares the error (e.g., root square error) between the element value of the tensor at the index of Ω at the time of the current update and the observed value with a threshold (which depends on the assumed magnitude of noise, but may be several percent of the maximum value of all observed values). If the error is equal to or less than the threshold, the consistency checking unit 25 retains the data as is, and if the error exceeds the threshold, adopts the observed value, or may correct the data by adopting a value obtained by adding or subtracting the allowable error from the observed value, whichever is closest to the currently updated value.

[0055] In the above-described embodiment, an example was given in which related factors were extracted from fluorescence analysis data, which is three-dimensional tensor data. However, the related factor extraction device of the present invention can also be applied to N-dimensional tensor data, which is four or more dimensions. When processing N-dimensional tensor data, this can be achieved by inductively performing the method described above. That is, the relevant parts of the processing of fluorescence analysis tensor data can be expanded as follows. The main points are explained below.

[0056] The factorization calculation unit 22 first takes the Nth orthonormal matrix and applies its transpose to the Nth dimension of the tensor. Next, the same operation is performed on the (N-1)th dimension. Generally, the jth orthonormal matrix is ​​taken and its transpose is applied to the jth dimension of the tensor. This is repeated until j=3, and when j=2, the matrix spanned by the first and second dimensions of the tensor is subjected to singular value decomposition. A unique factorization result is output from the input tensor and the above (N-2) orthonormal matrices.

[0057] To update the (N-2) orthonormal matrices, take an index where j>2, set it as j', fix all indices other than j', and update the j'th orthonormal matrix. This process starts from j=N, and for j>2, updates are performed by descending the j index. Then, the factorization using the current (N-2) orthonormal matrices updated by the factorization calculation unit can be obtained.

[0058] In this invention, we can provide factorization that can represent data with a small number of elements while complementing missing tensor data. It is expected that the data structure can be captured even from only partial observations. [Explanation of symbols]

[0059] 1. Related Factor Extraction Device 10 Data Acquisition Section 11 Arithmetic section 12 Storage section 13 Output section 21 Missing element specification section 22 Factorization calculation section 23 Reduction direction calculation section 24 Provisional Restoration Section 25 Consistency check section

Claims

1. A related factor extraction device that extracts related factors from observation data of a missing multidimensional tensor, comprising: a missing element designation unit that designates missing elements of the multidimensional tensor data; a factorization calculation unit that multiplies (the number of dimensions - 2) dimensions of the multidimensional tensor data by an orthonormal matrix to generate two-dimensional tensor data, and performs singular value decomposition on the two-dimensional tensor data to factorize the multidimensional tensor data; a reduction direction calculation unit that calculates a gradient direction that reduces a sum of coefficients of factorization for the orthonormal matrix and calculates a new orthonormal matrix in the gradient direction; a provisional reconstruction unit that reconstructs multidimensional tensor data using the orthogonal matrix and the singular value decomposition results calculated by the factorization calculation unit and the reduction direction calculation unit; a consistency checking unit that performs processing to align the multidimensional tensor data restored by the provisional restoration unit with the observation value data; Equipped with a factorization calculation unit that calculates the factorization result of the factorization calculation unit and the reduction direction calculation unit; a factorization calculation unit that calculates the factorization result of the factorization calculation unit and the reduction direction calculation unit; a factorization calculation unit that calculates the factorization result of the factorization calculation unit and the reduction direction calculation unit;

2. 2. The related factor extraction device according to claim 1, wherein the reduction direction calculation unit calculates a new orthonormal matrix by adding a vector indicating the gradient direction to a current orthonormal matrix and mapping the added matrix to a space in which orthonormal matrices exist.

3. 3. The related factor extraction device according to claim 2, wherein the reduction direction calculation unit uses a backtracking method to find a gradient direction in which a sum of coefficients of factorization is equal to or less than a predetermined threshold.

4. The related factor extraction device according to claim 1 , wherein the consistency checking unit replaces values ​​of the restored multidimensional tensor data with the observed value data for tensor elements for which the observed value data exists.

5. 4. The related factor extraction device according to claim 1, wherein the consistency checking unit replaces a tensor element for which the observed value data exists with the observed value when an error between the value of the restored multidimensional tensor data and the observed value is equal to or greater than a threshold.

6. 4. The related factor extraction device according to claim 1, wherein, for a tensor element for which the observed value data exists, if an error between a value of the restored multidimensional tensor data and the observed value data is equal to or greater than a threshold, the consistency checking unit replaces the tensor element with a value determined based on the observed value and the value of the multidimensional tensor data.

7. A method for extracting related factors from observation data of a missing multidimensional tensor, comprising: a first step of specifying missing elements of multidimensional tensor data; a second step of multiplying (the number of dimensions - 2) dimensions of the multidimensional tensor data by an orthonormal matrix to obtain two-dimensional tensor data, and factorizing the two-dimensional tensor data by singular value decomposition; a third step of calculating a gradient direction that reduces the sum of the factorization coefficients for the orthonormal matrix, and calculating a new orthonormal matrix in the gradient direction; a fourth step of restoring multidimensional tensor data using the orthonormal matrix and the result of singular value decomposition calculated in the second step and the third step; a fifth step of performing a process of matching the multidimensional tensor data restored in the fourth step with the observed value data; Equipped with a method for extracting associated factors, the method comprising: repeating the second step, the third step, the fourth step, and the fifth step using the restored data that has been subjected to processing for aligning with the observed value data in the fifth step until a predetermined termination condition is satisfied; and determining, when the predetermined termination condition is satisfied, the result of the factorization determined in the second step as the associated factors of the multidimensional tensor data.

8. A program for extracting relevant factors from observation data of a missing multidimensional tensor, the program comprising: a first step of specifying missing elements of multidimensional tensor data; a second step of multiplying (the number of dimensions - 2) dimensions of the multidimensional tensor data by an orthonormal matrix to obtain two-dimensional tensor data, and factorizing the two-dimensional tensor data by singular value decomposition; a third step of calculating a gradient direction that reduces the sum of the factorization coefficients for the orthonormal matrix, and calculating a new orthonormal matrix in the gradient direction; a fourth step of restoring multidimensional tensor data using the orthonormal matrix and the result of singular value decomposition calculated in the second step and the third step; a fifth step of performing a process of matching the multidimensional tensor data restored in the fourth step with the observed value data; Execute a program for repeating the second, third, fourth, and fifth steps using the restored data that has been processed to be consistent with the observed value data in the fifth step until a predetermined termination condition is satisfied, and for determining the result of the factorization determined in the second step as the associated factor of the multidimensional tensor data when the predetermined termination condition is satisfied.

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