JP2019079291A - Matrix decomposition device and matrix decomposition method - Google Patents

Abstract

To reduce loads of calculation for decomposing a real-number matrix.SOLUTION: A matrix decomposition device 10 that decomposes a real-number matrix W into a product of a basis matrix M composed of discrete values and a coefficient matrix C composed of real-numbers comprises: a vector extraction unit 12 that extracts a real-number vector b of each row of the real-number matrix W; a coefficient matrix determination unit 13 that determines the coefficient matrix C; an optimum solution search unit 14 that obtains an optimum solution vector m of each row of the basis matrix M corresponding to the real-number vector b of each extracted row using the determined coefficient matrix C; and a decomposition matrix output unit 17 that combines an optimum solution vector m of each row obtained by the optimum solution search unit 14 to output as the basis matrix M and that outputs the coefficient matrix C used in obtaining the basis matrix M. The optimum solution search unit 14 performs hybrid search repetitively switching Monte Carlo tree search and branch-and-bound search alternately while taking over a search tree and provisional minimum error.SELECTED DRAWING: Figure 1

Description

  The present invention relates to a matrix factorization apparatus and matrix factorization method that decomposes a matrix composed of real numbers into a product of a matrix having real numbers and a matrix having only integer numbers.

  The image recognition operation may include a product operation of a vector composed of real numbers (real vector) and a matrix composed of real numbers (real matrix). For example, SVM (Support Vector Machine) includes a product operation of a feature vector which is a real vector and a weighting matrix which is a real matrix, and a CNN (Convolutional Neural Network) has an input layer (vector) which is a real vector. And a convolution filter (matrix) which is a real matrix. The calculation load of the product calculation of such a real number vector and a real number matrix is large, that is, it takes a long calculation time, and the memory consumption becomes large.

  Therefore, conventionally, while converting a real vector to a vector having only integer values, the load of product operation is reduced by decomposing a real matrix into a product of a base matrix having only integer values and a coefficient matrix having real values. It has been proposed to do.

The problem of decomposing a real number matrix W into a coefficient matrix C and a basis matrix M can be reduced to the following equation (1).

  In order to find an optimal solution to this problem, basically it is necessary to search all solutions, but depending on the size of the problem, astronomical calculation time may be required. Therefore, there are several approaches as existing methods for solving this problem.

  First, since this problem can be regarded as a constrained optimization problem, it can be solved by the branch and bound method. The branch and bound method calculates the smallest unconstrained error that the remaining elements can realize as an unconstrained optimization problem in the process of determining all the elements. If the currently obtained minimum error (temporary minimum error) is smaller than this minimum unconstrained error, regardless of the remaining elements, no solution can be obtained that makes the error smaller than the temporary minimum error. It becomes possible to prun the search tree. The branch and bound method narrows the search range deterministically by such pruning, and outputs a combination that realizes the temporary minimum error at that time as a global optimum when there is no branch to be searched. .

  Moreover, there is a Monte Carlo tree search as a probabilistic approach (for example, Patent Document 1). The Monte Carlo tree search is an approach that seeks to obtain a probabilistically good solution by giving up searching for all solutions. Above all, it is known that a method called UCT (Upper Confidence Bounds for Trees) is applied in artificial intelligence or the like pointing to Go, and has high performance.

  In UCT, values are searched randomly at the beginning of the search, but when the search progresses, the expected value of the error obtained by selection from the history of the past errors is calculated, and it is small from 0 and 1 Search for a candidate that is likely to get an error. In addition, there is also a possibility that there is an element that can not be searched at all if an expected value alone results in an accidental small error. For this reason, the expected value is weighted such that an element with a small number of times selected in the past is preferentially selected.

JP, 2014-142849, A

  The branch and bound method has the advantage that the global optimum solution can be obtained strictly as described above, but there is a problem that the operation time becomes too large if the pruning does not work well. That is, in the branch and bound method, when a solution close to the global optimum solution is obtained early, there is a possibility that pruning can proceed efficiently, but otherwise the efficiency becomes extremely low.

  On the other hand, Monte Carlo tree search such as UCT has the advantage that a relatively good solution can be obtained in a short time, but there is no guarantee that a strictly global optimal solution can be obtained. There is a problem that it takes more computation time than that.

  The present invention has been made in view of the above problems, and an object thereof is to reduce the operation load for decomposing a real number matrix.

  One aspect of the present invention is a matrix factorization apparatus (10) for decomposing a real number matrix (W) into a product of a basis matrix (M) consisting of discrete values and a coefficient matrix (C) consisting of real numbers, W) determined by the vector extraction unit (12) for extracting the real vector (b) of each row, the coefficient matrix determination unit (13) for determining the coefficient matrix (C), and the coefficient matrix determination unit (13) The optimal solution vector (m) of each row of the basis matrix (M) corresponding to the real vector (b) of each row extracted by the vector extraction unit (12) is determined using the coefficient matrix (C) An optimal solution search unit (14) and the optimal solution vector (m) of each row obtained by the optimal solution search unit (14) are combined and output as the basis matrix (M), and the basis matrix (M) Output the coefficient matrix (C) used when obtaining A solution matrix output unit (17), wherein the optimum solution search unit (14) performs a hybrid search that alternately switches between Monte Carlo tree search and branch-and-bound search while taking over the search tree and the provisional minimum error. Have.

  According to this configuration, since a solution close to the optimal solution can be searched early by Monte Carlo tree search, pruning in branch limited search effectively progresses, and if pruning proceeds by branch limited search, it is more optimal in Monte Carlo tree search Since the synergetic effect that the solution close to the solution can be searched early is obtained, the calculation load can be reduced and the optimum solution can be effectively found.

  The above-mentioned matrix decomposition apparatus (10) is the above-mentioned decomposition error (E) when the above-mentioned optimum solution vector (m) of each row obtained by the above-mentioned optimum solution search unit (14) is combined to form the above-mentioned basis matrix (M). The apparatus may further include a decomposition error determination unit (16) that determines whether or not the value has become sufficiently small based on a predetermined standard, and the decomposition matrix output unit (17) includes the decomposition error determination unit (16). The base matrix (M) and the coefficient matrix (C) may be output when it is determined that the decomposition error (E) has become sufficiently small at step. When it is determined that the error (E) is not sufficiently small, the coefficient matrix determination unit (13) determines a new coefficient matrix (C), and the optimal solution search unit (14) New optimal solution vector using new coefficient matrix (C) It may be asked Le. With this configuration, the resolution error can be reduced to improve the resolution accuracy.

  In the above matrix decomposition apparatus (10), the optimum solution search unit (14) performs the hybrid search until a global optimum solution vector is obtained by the branch and bound search, and the global optimum solution vector is determined as the optimum solution May be determined as a vector. With this configuration, the resolution error can be reduced to improve the resolution accuracy.

  In the above-described matrix factorization device (10), the optimum solution search unit (14) may switch between the Monte Carlo tree search and the branch and bound search when updating of the temporary minimum error is delayed.

  In the above-described matrix decomposition apparatus (10), the optimum solution search unit (14) may switch between the Monte Carlo tree search and the branch-limited search when a predetermined time has elapsed since the previous switching.

  In the above-described matrix decomposition apparatus (10), the optimal solution search unit (14) may switch between the Monte Carlo tree search and the branch and bound search when an error is calculated a predetermined number of times.

  In the above-described matrix factorization apparatus (10), when the predetermined time has elapsed since the previous switching or the optimum solution search unit (14) calculates an error a predetermined number of times, the Monte Carlo tree search method and the branch limitation You may switch between searching and searching.

  One aspect of the present invention is a matrix decomposition method for decomposing a real number matrix (W) into a product of a basis matrix (M) consisting of discrete values and a coefficient matrix (C) consisting of real numbers, and the real number matrix (W) Acquiring, extracting a real number vector (b) of each row of the real number matrix (W), determining a coefficient matrix (C), the real number vector (b), and the coefficient matrix (C) Determining an optimal solution vector (m) of each row of the basis matrix (M) for each real vector (b) of each row using all the real vectors (b) extracted from the real matrix (W) And combining the optimal solution vector (m) obtained for the matrix and outputting the combined matrix as the basis matrix (M), and outputting the coefficient matrix (C) used when the basis matrix (M) is obtained. The optimal solution vector (m Step Repeated hybrid search to switch alternately and Monte Carlo tree search and branch and bound search while taking over the search tree and temporary minimum error, matrix decomposition method for obtaining the.

  According to this configuration, since a solution close to the optimal solution can be searched early by Monte Carlo tree search, pruning in branch limited search effectively progresses, and if pruning proceeds by branch limited search, it is more optimal in Monte Carlo tree search Since the synergetic effect that the solution close to the solution can be searched early is obtained, the calculation load can be reduced and the optimum solution can be effectively found.

  According to the present invention, since a solution close to the optimal solution can be searched early by Monte Carlo tree search, pruning in branch limited search effectively proceeds, and if pruning proceeds by branch limited search, Monte Carlo tree search Since the synergetic effect of being able to search for a solution closer to the optimum solution earlier can be obtained, the calculation load can be reduced and the optimum solution can be effectively found.

The figure which shows the structure of the row example decomposition | disassembly apparatus of embodiment of this invention. Diagram showing decomposition of real number matrix according to the embodiment of the present invention A diagram showing a search tree used in the optimum solution search unit of the embodiment of the present invention Flow chart of the matrix decomposition method executed by the matrix decomposition apparatus according to the embodiment of the present invention Flow chart of the optimal solution search process executed by the matrix factorization apparatus according to the embodiment of the present invention

  Hereinafter, embodiments of the present invention will be described with reference to the drawings. The embodiment described below shows an example in the case of practicing the present invention, and the present invention is not limited to the specific configuration described below. In the implementation of the present invention, a specific configuration according to the embodiment may be adopted as appropriate.

  FIG. 1 is a diagram showing the configuration of a row decomposition apparatus according to an embodiment of the present invention. The configuration shown in FIG. 1 is realized by the CPU of a computer including a CPU, a memory, a storage device, and the like executing the matrix decomposition program according to the present embodiment. The matrix division apparatus 10 according to the present embodiment includes a real number matrix input unit 11, a vector extraction unit 12, a coefficient matrix determination unit 13, an optimal solution search unit 14, a switching determination unit 15, and a decomposition error determination unit 16 , And a decomposition matrix output unit 17. The optimum solution search unit 14 includes a Monte Carlo tree search unit 141 and a branch-and-bound search unit 142.

  FIG. 2 is a diagram showing decomposition of a real number matrix. The matrix factorization unit 10 inputs a real matrix W of D rows and L columns as shown in FIG. 2 and inputs it into a base matrix M consisting of discrete values of D rows and K columns (K is the number of bases), K rows L It is decomposed into the product of the column with the coefficient matrix C, and the obtained basis matrix M and the coefficient matrix C are output. In this embodiment, the basis matrix M is a binary matrix consisting of −1 and 1, but even if the basis matrix M is a ternary matrix consisting of −1, 0 and 1, it can be selected from any integer The matrix may be

  The real number matrix input unit 11 obtains a real number matrix W to be decomposed that is externally input. The vector extraction unit 12 extracts the transposed vector of the row vector of each row of the input real number matrix W as a column vector (real number vector) b and outputs the row to the optimum solution search unit 14 for each row. The coefficient matrix determination unit 13 determines the coefficient matrix C and outputs the coefficient matrix C to the optimum solution search unit 14.

The optimal solution search unit 14 searches for the optimal basis matrix M, using the coefficient matrix C as a known one given from the coefficient matrix determination unit 13. The optimum solution search unit 14 obtains an optimum solution at high speed by a hybrid search that alternately switches between a Monte Carlo tree search and a search by branch and bound method (hereinafter referred to as “branch and bound search”) alternately. As described above, the problem of decomposing a real number matrix W into a coefficient matrix C and a basis matrix M can be reduced to equation (1).

である。本実施の形態において一回の計算で求める基底行列Ｍの要素は、図２に示す基底行列Ｍの行ベクトルｍである。 In this equation (1), the cost function is

It is. The elements of the basis matrix M to be obtained by one calculation in the present embodiment are the row vectors m of the basis matrix M shown in FIG.

The optimal solution search unit 14 performs a search for each row of the basis matrix M. In the case of the row vector m, since the notation becomes complicated, the transposed vector of the row vector m of each row of the basis matrix M is redefined as the column vector x in the following description as follows.

となり、行ごとの最適解は、

と再定義される。 Redefining as above, the cost function is

And the optimal solution for each row is

Redefined as

  FIG. 3 is a diagram showing a search tree used in the optimum solution search unit 14. In the present embodiment, since the basis matrix M is a binary matrix, a binary tree is used as a search tree. FIG. 3 shows a binary tree in the case of the basis number K = 4 as an example. In FIG. 3, the k-th node at depth corresponds to whether the value of x k is -1 or 1. If the binary tree is traced from the root node at the top to the leaf nodes at the bottom, values of x1 to xK are given (in the example of FIG. 3, values of x1 to x4 are given), and the error at that time is calculated it can.

  As described above, the optimum solution search unit 14 obtains the optimum solution vector by the hybrid search that alternately switches the Monte Carlo tree search and the branch-and-bound search alternately. That is, the optimal solution search unit 14 alternates between the Monte Carlo tree search and the branch limited search while inheriting the provisional minimum error and the vector x (provisional optimum solution vector) when the provisional minimum error is obtained, and the pruning information. Switch to and find the optimal solution vector. In other words, the Monte Carlo search unit 141 and the branch only search unit 142 share the binary tree and the provisional minimum error and perform the search. In the following, the Monte Carlo tree search in the Monte Carlo search unit 141, the branch only search in the branch only search unit 142, and switching between them will be described.

(Monte Carlo tree search)
The Monte Carlo tree search unit 141 searches for a path with a small error probability from among paths of 2 4 powers (column size of the base matrix M) from a root node of a binary tree which is a search tree to leaf nodes. Each node holds the score s and the number of visits n as values. One search ends when it reaches the leaf node for the first time from the root node.

  All nodes other than leaf nodes have two child nodes. When focusing on a certain node, if both child nodes of the focused node have not been traced yet, the Monte Carlo tree search unit 141 randomly selects with an equal probability from among those two child nodes. Decide which node to follow next. If only one of the two child nodes has been visited, the Monte Carlo tree search unit 141 determines the node that has not been visited as the next node.

  When both child nodes are pruned by the branch and bound search described later, the Monte Carlo tree search unit 141 returns the route to the root node. When only one child node is pruned by the branch and bound search, the Monte Carlo tree search unit 141 sets the remaining child node as the next node.

ここで、ｔは、それまでに探索したリーフノードの数である。 If both child nodes have been traced, the Monte Carlo tree search unit 141 compares the UBC scores of the two child nodes and determines the larger one as the next node. The UBC score is calculated by the following equation.

Here, t is the number of leaf nodes searched so far.

The Monte Carlo tree searching unit 141 calculates an error e when arriving from the root node to the leaf node according to the method of determining child nodes in each of the above nodes. The Monte Carlo tree search unit 141 further converts this error e into a leaf score l according to the following equation.

  Further, the Monte Carlo tree searching unit 141 adds the leaf score l to the score s and adds 1 to the number of visits n for all the nodes on the route from the root node to the leaf note reached. Then, the Monte Carlo tree searching unit 141 compares the obtained error e with the temporary minimum error, and updates and stores the obtained error e as a temporary minimum error if the temporary minimum error is larger. Further, the Monte Carlo tree search unit 141 stores the solution at that time (that is, the vector x formed of a combination of the paths x1 to xK for which the provisional minimum error is calculated) as a provisional optimal solution vector.

  When an error e is calculated at a leaf node, that leaf node is pruned. Also, in each node, when both of its two child nodes are pruned, the node itself is pruned.

  The Monte Carlo tree search unit 141 repeats this search to update the provisional minimum error. In the Monte Carlo tree search, the search ends when there are no more nodes to be searched, but generally, the size of the search tree is astronomically large, so this situation does not occur, and a predetermined When the condition is satisfied, the search is aborted and the temporary optimal solution at that time is output.

(Branch only search)
The branch and bound search unit 142 searches for an optimal solution using a branch and bound method. The branch and bound search unit 142 prunes nodes that do not need to be logically traced out of a path according to the power of 2 4 (the column size of the base matrix M) from the root node to the leaf nodes to reduce the search range While searching for the global optimum solution.

  Focusing on a certain node, the branch-boundary search unit 142 determines the left child node as the next node if the left child node of the two child nodes of the focused node has not been pruned yet Do. In addition, when the branch and bound search unit 142 reaches the leaf node and calculates the error e, it prunes the leaf node. If the two child nodes of the node of interest are both pruned, the branch-and-bound search unit 142 prunes the node of interest itself.

Now, when focusing on a node having a depth k, the branch and bound search unit 142 arranges values of x1 to xk determined by a route traced from the root node to the node of interest, and creates a partial problem as follows. .

The branch and bound search unit 142 solves this subproblem as a new problem for obtaining a real vector xf without discrete constraints as follows.

  The branch and bound search unit 142 finds the minimum value by solving this subproblem using a least squares method or the like. If this minimum value is larger than the provisional minimum error, the branch and bound search unit 142 prunes this node. This is because, since xf is a real vector, it is not possible to obtain a solution with a smaller error than this minimum value by tracing any optimal path in the search tree below this node.

  When the focused node is a leaf node, the row of -1 and 1 from the root node to the leaf node is a candidate for a solution, so the branch and only searching unit 142 determines xk from the root node to the leaf node. Arrange the values of to create a vector and calculate its error e. If the error e is smaller than the tentative minimum error, the branch and bound search unit 142 updates and stores the tentative minimum error with the error e. In addition, the branch and bound search unit 142 stores the solution at that time (that is, the vector x formed of a combination of the paths x1 to xK for which the provisional minimum error is calculated) as a provisional optimal solution vector.

  When all the nodes are pruned, the branch and bound search unit 142 outputs the temporary optimal solution at that time to the decomposition error determination unit 16 as a global optimal solution.

  Next, switching between the Monte Carlo tree search by the Monte Carlo tree search unit 141 and the branch only search by the branch only search unit 142 will be described. In the Monte Carlo tree search in the Monte Carlo tree search unit 141, the switching determination unit 15 determines whether or not the switching condition from the Monte Carlo tree search to the branch-limited search is satisfied each time the leaf node is reached or periodically. . When the switching solution determination unit 15 receives the determination result that the switching determination unit 15 satisfies the switching condition, the optimum solution searching unit 14 ends the process by the Monte Carlo tree searching unit 141 and starts the process by the branch and limitation searching unit 142.

  The switching determination unit 15 determines whether or not the switching condition from the branch only search to the Monte Carlo tree search is satisfied every time the branch node reaches the leaf node or periodically in the branch only search in the branch only search unit 142. judge. When the switching solution determination unit 15 receives the determination result that the switching determination unit 15 satisfies the switching condition, the optimum solution searching unit 14 ends the process by the branch and limitation searching unit 142 and starts the process by the Monte Carlo tree searching unit 141. Further, as described above, when all the nodes are pruned, the branch and bound search unit 142 outputs the temporary optimal solution vector (global optimal solution vector) at that time to the basis matrix update unit 16.

  The switching condition from the Monte Carlo tree search to the branch limited search in the switching determination unit 15 and the switching condition from the branch limited search to the Monte Carlo tree search may be the same or different. A first example of the switching condition in the switching determination unit 15 is that updating of the temporary minimum error is delayed. Specifically, when the switching determination unit 15 calculates an error a predetermined number of times (that is, the leaf node is reached a predetermined number of times), the temporary minimum error is not updated more than the predetermined number of times (may be one). It is determined that the switching condition is satisfied, assuming that the update of the temporary minimum error is delayed.

  A second example of the switching condition is that a predetermined time has elapsed since the previous switching. In this example, the switching determination unit 15 determines that the switching condition is satisfied when a predetermined time has elapsed after switching from Monte Carlo tree search to branch limited search, and switches from branch limited search to Monte Carlo tree search. When a predetermined time passes after switching from the branch-limited search to the Monte Carlo tree search, it is determined that the switching condition is satisfied, and the branch-and-bound method is switched to the Monte Carlo tree search.

  A third example of the switching condition is that the error has been calculated a predetermined number of times (that is, the leaf node has reached a predetermined number of times). A fourth example of the switching condition is to satisfy either the predetermined time elapses from the previous switching (second example) or the error is calculated a predetermined number of times or more (third example).

  Further, the switching determination unit 15 may randomly perform switching determination. In this case, the switching probability may be determined arbitrarily. The switching determination unit 15 may randomly determine switching after calculating the error a predetermined number of times.

  The vector extraction unit 12, the optimal solution search unit 14, and the switching determination unit 15 perform the above-described process on all the row vectors w 1 to w D of the real number matrix W. As a result, the decomposition error determination unit 16 obtains D global optimum solutions m1 to mD corresponding to the D row vectors w1 to wD.

  The decomposition error determination unit 16 generates the basis matrix M by arranging the global optimum solutions m1 to mD in order, and using the basis matrix M and the coefficient matrix C for which it is calculated, the real number matrix W and the basis matrix M The difference between the product of and the coefficient matrix C is determined as a decomposition error E. The decomposition error judgment unit 16 judges whether the decomposition error E is smaller than the provisional minimum decomposition error, and updates the provisional minimum decomposition error with the decomposition error E if the decomposition error E is smaller than the provisional minimum decomposition error. Then, the basis matrix M and the coefficient matrix C given the decomposition error E are stored.

  The decomposition error determination unit 16 outputs a basis matrix M generated by arranging the global optimum solutions m1 to mD in order to the coefficient matrix determination unit 13. The coefficient matrix determination unit 13 determines (updates) the coefficient matrix C by the least squares method based on the real number matrix W and the basis matrix M. Note that the coefficient matrix determination unit 13 determines (initializes) the coefficient matrix C with random numbers since the basis matrix M is not obtained at the beginning of the process in which the process in the optimum solution search unit 14 is not yet performed. To the optimum solution search unit 14.

  When the provisional minimum decomposition error in the decomposition error determination unit 16 becomes sufficiently small, the decomposition matrix output unit 17 outputs the basis matrix M and the coefficient matrix C given the provisional minimum decomposition error as a result of matrix decomposition. . For example, when the provisional minimum decomposition error becomes equal to or less than a predetermined threshold, the decomposition matrix output unit 17 may determine that the provisional minimum decomposition error has become sufficiently small, and the provisional minimum decomposition error is not updated. It may be determined that the provisional minimum decomposition error has become sufficiently small, and the optimal solution search unit 14 executes the hybrid search a predetermined number of times (that is, when the coefficient matrix C is updated a predetermined number of times). It may be determined that the minimum resolution error has become sufficiently small.

  FIG. 4 is a flowchart of the matrix factorization method executed by the matrix factorization device of the present embodiment. The matrix factorization apparatus 10 first inputs the real number matrix W from the outside at the real number matrix input unit 11 (step S11). In addition, the coefficient matrix determination unit 13 randomly initializes the coefficient matrix C using random numbers (step S12). The vector extraction unit 12 and the optimum solution search unit 14 set i = 1 (step S13), and search for the optimum solution vector x for the column vector b (step S14).

  When obtaining the optimum solution vector x for the column vector b, the vector extraction unit 12 and the optimum solution search unit 14 determine whether i = D, that is, obtain the optimum solution vector x for all column vectors b of the real matrix W Is determined (step S15).

  If there is a column vector b for which the optimal solution vector x has not yet been obtained (NO in step S15), i is incremented (step S16), and the process returns to step S14 and the optimal solution vector for the next column vector b Search x (step S14). When optimal solution vectors x are obtained for all column vectors b of real number matrix W (YES in step S15), decomposition error determination unit 16 generates basis matrix M using those optimal solution vectors x, and The decomposition error E is calculated using the coefficient matrix C at that time (step S17).

  The decomposition error determination unit 16 determines whether to update the provisional minimum decomposition error based on whether the calculated decomposition error E is smaller than the provisional minimum decomposition error (step S18). When the calculated decomposition error E is smaller than the provisional minimum decomposition error (YES in step S18), the decomposition error determination unit 16 updates the provisional minimum decomposition error with the decomposition error E (step S19), and the calculated decomposition If the error E is larger than the provisional minimum decomposition error (NO in step S18), it is determined whether the decomposition is ended without updating the provisional minimum error with the decomposition error E (step S20). The determination of the end of decomposition depends on whether or not the provisional minimum decomposition error has become sufficiently small, as described above.

  When the decomposition is not completed (NO in step S20), coefficient matrix determination unit 13 updates coefficient matrix C using generated basic matrix M and input real number matrix W, and returns to step S13. Perform matrix decomposition again.

  When it is determined that the decomposition error determination unit 16 ends the decomposition (YES in step S20), the matrix output unit 17 uses the base matrix M and the coefficient matrix C from which the provisional minimum decomposition error is obtained as the decomposition result. It outputs (step S21).

  FIG. 5 is a flowchart of a subroutine of search processing (step S14) for the optimum solution vector x by the vector extraction unit 12 and the optimum solution search unit 14. The vector extraction unit 12 first extracts the transposed vector of the row vector of the ith row of the input real number matrix W as a column vector b (step S41). The Monte Carlo tree search unit 141 executes a Monte Carlo tree search using the column vector b and the coefficient matrix C (Step S42).

  The switch determination unit 15 determines whether the switch condition is satisfied (step S43), and if the switch condition is not satisfied (NO in step S43), the Monte Carlo tree search unit 141 continues the Monte Carlo tree search. (Step S42). When the switching determination unit 15 determines that the switching condition is satisfied (YES in step S43), the optimum solution searching unit 14 inherits the binary tree at that time, the provisional minimum error, and the provisional optimum solution vector, and searches the Monte Carlo tree. Is switched to the branch only search by the branch only search unit 142, and the branch only search unit 142 executes the branch only search (step S44).

  The switching determination unit 15 determines whether or not the switching condition is satisfied (step S45), and if the switching condition is not satisfied (NO in step S45), the branch limitation search unit 142 performs branch limitation search (Step S44). If the switching determination unit 15 determines that the switching condition is satisfied (YES in step S45), the optimum solution searching unit 14 determines whether the global optimum solution is obtained by the branch and only searching unit 142 (pruned) It is determined whether all the nodes that have not been traced) (step S46).

  If the global optimum solution has not been obtained yet (NO at step S46), the optimum solution search unit 14 inherits the binary tree at that time, the provisional minimum error and the provisional optimum solution vector, and Switching to the Monte Carlo tree search by the Monte Carlo tree search unit 141, the Monte Carlo tree search unit 141 executes the Monte Carlo tree search (step S42).

  Thus, the optimal solution search unit 14 alternately repeats the Monte Carlo tree search and the branch-limited search until the global optimum solution is obtained by the branch-limited search in the branch-limited search unit 142. Then, when the global optimum solution is obtained (YES in step S46), the optimum solution search unit 14 returns to the flow of FIG.

  As described above, the matrix factorization device 10 according to the present embodiment first fixes the coefficient matrix C in order to decompose the real number matrix W into the product of the basis matrix M consisting of discrete values and the coefficient matrix C consisting of real numbers. Then, the base matrix M is updated, and when the base matrix M is updated, the coefficient matrix C is updated accordingly, and the update of the base matrix M and the update of the coefficient matrix C are repeated until the decomposition error becomes sufficiently small.

  Then, when updating the basis matrix M by fixing the coefficient matrix C and fixing the coefficient matrix C, the matrix factorization device 10 of this embodiment performs a hybrid search that alternately repeats the Monte Carlo tree search and the branch-and-bound search for each row. Do. In this hybrid search, since a Monte Carlo tree search can provide a relatively good solution in a short time, a branch-and-bound search can quickly find a strictly global optimum solution. That is, according to the matrix factorization device 10 of the present embodiment, the speed and accuracy of matrix factorization can be improved.

  In the example of FIG. 5, the optimal solution search unit 14 outputs the global optimal solution vector obtained by the branch and bound method as the optimal solution vector for the column vector b. That is, although the optimum solution search unit 14 performs the hybrid search of the Monte Carlo tree search and the branch-and-bound search until the global optimum solution is obtained, the hybrid search may be aborted before the global optimum solution is obtained. For example, when switching between the Monte Carlo tree search (step S42) and the branch and bound method (step S44) is performed a predetermined number of times, the solution giving the temporary minimum error at that time is output as the optimum solution vector and the hybrid search is ended. Alternatively, when the provisional minimum error falls below a predetermined threshold, a solution giving the provisional minimum error may be output as an optimal solution vector to end the hybrid search.

  The present invention can reduce the operation load and efficiently find the optimum solution, and is useful as a matrix factorization device that decomposes a real number matrix into a product of a basis matrix consisting of discrete values and a coefficient matrix consisting of real numbers. .

DESCRIPTION OF SYMBOLS 10 matrix decomposition apparatus 11 real number matrix input part 12 vector extraction part 13 coefficient matrix determination part 14 optimal solution search part 141 Monte Carlo tree search part 142 branch limited search part 15 switch determination part 16 decomposition error determination part 17 decomposition matrix output part

Claims (8)

A matrix factorization apparatus (10) which decomposes a real number matrix (W) into a product of a basis matrix (M) of discrete values and a coefficient matrix (C) of real numbers,
A vector extraction unit (12) for extracting a real vector (b) of each row of the real matrix (W);
A coefficient matrix determination unit (13) that determines a coefficient matrix (C);
A base matrix (M corresponding to the real vector (b) of each row extracted by the vector extraction unit (12) using the coefficient matrix (C) determined by the coefficient matrix determination unit (13) An optimal solution search unit (14) for determining an optimal solution vector (m) of each row of
The coefficients used when the optimal solution vector (m) of each row determined by the optimal solution search unit (14) is combined and output as the basis matrix (M), and the basis matrix (M) is obtained A decomposition matrix output unit (17) that outputs a matrix (C);
Equipped with
The optimal solution search unit (14) is a matrix decomposition apparatus (10) that performs a hybrid search that alternately switches between Monte Carlo tree search and branch-and-bound search while taking over the search tree and the temporary minimum error. The decomposition error (E) when the optimal solution vector (m) of each row determined by the optimal solution search unit (14) is combined to form the base matrix (M) is sufficiently based on a predetermined criterion. The apparatus further comprises a decomposition error determination unit (16) that determines whether or not it has become smaller
The decomposition matrix output unit (17) is configured to determine the basis matrix (M) and the coefficient matrix (C) when the decomposition error determination unit (16) determines that the decomposition error (E) has become sufficiently small. Output),
The coefficient matrix determination unit (13) determines a new coefficient matrix (C) when the decomposition error determination unit (16) determines that the decomposition error (E) is not sufficiently small. The matrix decomposition apparatus (10) according to claim 1, wherein the optimum solution search unit (14) obtains a new optimum solution vector using the new coefficient matrix (C).   The optimal solution search unit (14) performs the hybrid search until a global optimal solution vector is obtained in the branch-and-bound search, and obtains the global optimal solution vector as the optimal solution vector. The matrix factorization device (10) as described.   The matrix decomposition apparatus according to any one of claims 1 to 3, wherein the optimum solution search unit (14) switches between the Monte Carlo tree search and the branch-and-bound search when updating of the temporary minimum error is delayed. (10).   The matrix decomposition apparatus according to any one of claims 1 to 3, wherein the optimal solution search unit (14) switches between the Monte Carlo tree search and the branch-and-bound search when a predetermined time has elapsed since the previous switching. (10).   The matrix decomposition apparatus (10) according to any one of claims 1 to 3, wherein the optimal solution search unit (14) switches between the Monte Carlo tree search and the branch-and-bound search when an error is calculated a predetermined number of times. .   The optimal solution search unit (14) switches between the Monte Carlo tree search method and the branch-limited search when a predetermined time has elapsed since the previous switching or when an error is calculated a predetermined number of times. The matrix factorization device (10) according to any one of the above. A matrix decomposition method for decomposing a real matrix (W) into a product of a basis matrix (M) of discrete values and a coefficient matrix (C) of real numbers,
Obtaining a real matrix (W);
Extracting a real vector (b) of each row of the real matrix (W);
Determining a coefficient matrix (C);
Determining an optimal solution vector (m) of each row of the basis matrix (M) for the real vector (b) of each row using the real vector (b) and the coefficient matrix (C);
The optimal solution vectors (m) determined for all the real vectors (b) extracted from the real matrix (W) are combined and output as the basis matrix (M) to obtain the basis matrix (M) Outputting the coefficient matrix (C) used in
Including
The step of obtaining the optimum solution vector (m) is a matrix decomposition method in which a hybrid search in which a Monte Carlo tree search and a branch-and-bound search are alternately switched repeatedly while taking over a search tree and a provisional minimum error is performed.

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