JP6058065B2 - Tensor data calculation device, tensor data calculation method, and program - Google Patents

Description

  The present invention relates to a factorization technique for extracting a factor pattern from a plurality of attribute information, and particularly relates to a non-negative tensor factorization technique.

  In general, log data such as purchase logs and check-in logs can be expressed as tensors. Further, since such data is expressed as a positive real value, the data expressed as a tensor can be subjected to factor analysis using a non-negative tensor factorization method. However, factorization technology is widely used as a log data analysis method, but has a problem that it takes a lot of calculation time. For example, Non-Patent Document 1 describes a technique regarding a general factorization technique.

  Consider an example where a user purchases a plurality of products and such data exists for several days. Generally, it is expressed as third-order tensor data of “user x product x day”, and is decomposed into R (Rank) bases. At this time, assuming that the number of users (I), the number of products (J), and the number of days (K), the tensor is expressed as a tensor of size (IxJxK), and the calculation amount of factorization is proportional to (IxJxKxR). For example, when 1000 users make purchases from 1000 types of items, such data is accumulated for 1000 days and decomposed into 100 bases, the amount of calculation is 100 billion times per iteration. You have to do scale computations. Assuming that iterative calculation is normally performed about 1000 times, arithmetic processing is performed on a scale of 100 trillion times, and the calculation takes several days when a recent general-purpose computer is used.

  Further, the memory area for holding the tensor data requires an order amount of (IxJxK), and requires several GB of memory even in the above-described scale. Therefore, physical calculation cannot be performed with large-scale data (increase in the number of users and the number of products).

  Here, a calculation example of non-negative tensor factorization will be described. Non-negative tensor factorization decomposes tensor data into tensor products of factor matrices while maintaining non-negativeness. For example, as shown in FIG. 1, the third-order tensor data X of “user [I] x product [J] x day [K]” can be decomposed into three factor matrices A, B, and C. It can be expressed as

なお、本明細書のテキストでは、便宜上、テンソル積の記号を＊と表すことにする。また、本明細書のテキストでは、推定量を示すハット"＾"の記号を、便宜上、文字の頭上でなく、文字の直前に記載する。例えば、"＾Ｘ"のようにである。

In the text of this specification, the symbol of the tensor product is represented by * for convenience. Moreover, in the text of this specification, the symbol of the hat “＾” indicating the estimated amount is described immediately before the character, for the sake of convenience. For example, “^ X”.

  The above A, B, and C are non-negative matrices of IxR, JxR, and KxR, respectively, and the tensor product * is represented by the product of each base as follows.

上記のテンソルデータＸと、Ａ＊Ｂ＊Ｃが近似的に等しくなるように、Ａ、Ｂ、Ｃを求める手法がテンソル因子分解である。この時、＾Ｘ＝Ａ＊Ｂ＊Ｃとして、Ｄ（Ｘ｜｜＾Ｘ）を最小化させる。Ｄ（・）は距離関数であり、一般化ＫＬダイバージェンス（ｇＫＬ）距離では以下のように表される。

A technique for obtaining A, B, and C so that the above tensor data X and A * B * C are approximately equal is tensor factorization. At this time, D (X || ^ X) is minimized as ^ X = A * B * C. D (•) is a distance function, and is expressed as follows in the generalized KL divergence (gKL) distance.

距離関数をｇＫＬとした時、因子分解の最適値を推定するための更新式は下記のようになる。

When the distance function is gKL, the update formula for estimating the optimum factorization value is as follows.

更新式を何回か繰り返し適用することで、因子分解後のＡ、Ｂ、Ｃが得られる。図２は、例として因子行列Ａの更新処理手順を示している。図２に示すように、ａ_ｉｒの更新では、ａ_ｉｒ自体についてＩｘＲのループ処理が必要となり、ａ_ｉｒの値を求めるためにＪｘＫのループ処理が必要となる。従って、最終的にＩｘＪｘＫｘＲ回のループ処理が必要になる。また、＾Ｘ_ｉｊｋ、ｂ_ｊｋ、ｃ_ｋｒの値を求めるためにも同様の処理回数が要求される。

A, B, and C after factorization are obtained by repeatedly applying the update formula several times. FIG. 2 shows an update process procedure of the factor matrix A as an example. As shown in FIG. _2, the updating of _{a ir,} loop processing of IxR for _{a ir} itself is _required, the loop process of JxK to determine the value of _{a ir} is required. Therefore, IxJxKxR loop processing is finally required. In addition, the same number of processes is required to obtain the values of ＸX _ijk , b _jk , and c _kr .

Liu, Weixiang, Tianfu Wang, and Siping Chen. "Nonnegative tensor factorization for clustering genes with time series microarrays from different conditions: A case study." Biomedical Engineering and Informatics (BMEI), 2010 3rd International Conference on. Vol. 6. IEEE , 2010.

  In general, many tensor data have many elements that do not have a value, and the data structure is often sparse (the sparse is used here as a sparse expression). For example, in a purchase log, not all users purchase all commodities, and the tensor element is often treated as a meaningless value as “0”. As a specific example, there is a “missing value sparse tensor” that treats as a missing value when the data collection period varies from user to user or when some of the collected data is missing, and the user does not intentionally purchase There is a “zero-value sparse tensor” that treats as zero values (such as men not buying women's underwear), and all of them omit the elements that do not need to be calculated in the factorization update formula, and only the elements that need to be calculated Calculation can be performed at high speed by performing arithmetic processing.

  For example, the update formula of the missing value sparse tensor when the number of logs is L is as follows, and the update procedure of the factor matrix A is as shown in FIG.

図３に示すように、ａ_ｉｒの更新では、ａ_ｉｒ自体に対するＩｘＲのループ処理が必要ではあるが、ａ_ｉｒの値を求めるためにＪｘＫのループ処理は行わず、ｉに対するｊ，ｋのログ数Ｌ_ｉの処理回数でよく、最終的にＬｘＲ回のループ処理でよい。＾Ｘ_ｉｊｋ、ｂ_ｊｋ、ｃ_ｋｒの値を求めるためにも同様の処理回数でよく、スパースなデータ構造では計算の処理回数はＬｘＲ回に比例する。よって、計算量を大幅に削減することが可能である。また、物理的なメモリ量も（Ｌ）のオーダとなり、汎用ＰＣによる処理が可能となる。

As shown in FIG. _3, the updating of _{a ir, a ir} loop processing IxR against itself is required. _However, the loop process of JxK to determine the value of _{a ir} is not performed, j for i, k log well treatment times of a few L _i, finally may loop processing of LxR times. In order to obtain the values of ^ X _ijk , b _jk , and c _kr , the same number of processes may be used. In a sparse data structure, the number of calculations is proportional to LxR. Therefore, it is possible to greatly reduce the calculation amount. Also, the physical memory amount is on the order of (L), and processing by a general-purpose PC is possible.

  Further, for example, the update formula of the zero-value sparse tensor is as follows, and the update procedure of the factor matrix A is as shown in FIG.

ここで、図４に示すように、０値スパーステンソルの更新手順においては、分母を分子とは別のループで処理をしている。すなわち、ａ_ｉｒの更新では、「数８」に示すように分母はｉに非依存な値のため、分母をｉに関してのＩ回のループ処理の外に出すことができる。分母はｒに依存するため、ｒ個のデータ配列で取得しておくことによって分母の計算量は（Ｊ＋Ｋ）ｘＲ回に比例する程度に抑えることが可能である。ｂ_ｊｋ、ｃ_ｋｒの値を求めるためにも同様に、分母と分子を切り分けて計算することが可能であり、その結果、分母の計算量が（（Ｉ＋Ｊ＋Ｋ）ｘＲ）のオーダになり、分子の計算量が（ＬｘＲ）のオーダになる。しかしながら、一般的にはＩ，Ｊ，Ｋはログ数Ｌと比較して十分小さな値になるため、全体の計算量に対して分母の計算量は無視できるオーダになり、欠損値スパーステンソルの計算量と同程度のものになる。また、物理的なメモリ量も（Ｌ）のオーダとなり、汎用ＰＣによる処理が可能である。

Here, as shown in FIG. 4, in the update procedure of the zero-value sparse tensor, the denominator is processed in a loop different from the numerator. That is, in the update of a _ir , since the denominator is a value independent of i as shown in “ _Equation 8”, the denominator can be out of I loop processing for i. Since the denominator depends on r, the calculation amount of the denominator can be suppressed to be proportional to (J + K) × R times by obtaining r data arrays. Similarly, in order to obtain the values of b _jk and c _kr , the denominator and the numerator can be calculated separately. As a result, the calculation amount of the denominator is on the order of ((I + J + K) xR), The calculation amount is on the order of (LxR). However, in general, I, J, and K are sufficiently small values as compared with the number of logs L, so that the calculation amount of the denominator is negligible with respect to the total calculation amount, and the calculation of the missing value sparse tensor is performed. It will be about the same amount. Also, the physical memory amount is on the order of (L), and processing by a general-purpose PC is possible.

  However, in any case, the sparse non-negative tensor factorization can reduce the amount of calculation in principle, but unlike the matrix factorization method, it is an effective acceleration method due to the complexity of the data structure. Is not established. In particular, if the data structure is simply held, random access to the memory occurs, and processing is slow in large-scale processing.

  The present invention has been made in view of the above points, and provides a technique for speeding up the processing for factoring the sparse tensor and reducing the amount of memory and the amount of memory access communication required for the processing. The purpose is to do.

According to an embodiment of the present invention, a tensor data calculation device that performs loop calculation on N-dimensional (N is an integer of 3 or more) tensor data for a plurality of indexes of the tensor data,
The tensor data elements are ordered in the order of the loop direction of each index in the order of each index from the lowest index of the calculation loop to the previous highest index, and in the tensor data according to the ordering. For a non-empty element, the value of the element and the index value of the index from the lowest index indicating the position on the tensor data of the element to the immediately preceding index are arranged on the data storage unit, and Data placement processing means for calculating the number of non-empty elements and placing the data storage unit in the data storage unit, for each order of the highest index in the calculation loop, the number of the order of the index,
There is provided a tensor data calculation device comprising: calculation processing means for performing loop calculation on tensor data arranged on the data storage unit by the data arrangement processing means.

According to the embodiment of the present invention, a tensor data calculation method executed by a tensor data calculation device that performs loop calculation for a plurality of indexes of tensor data for N-dimensional (N is an integer of 3 or more) tensor data. There,
The tensor data elements are ordered in the order of the loop direction of each index in the order of each index from the lowest index of the calculation loop to the previous highest index, and in the tensor data according to the ordering. For a non-empty element, the value of the element and the index value of the index from the lowest index indicating the position on the tensor data of the element to the immediately preceding index are arranged on the data storage unit, and A data placement processing step for calculating the number of non-empty elements and placing the data storage unit in the data storage unit for each order of the highest index in the calculation loop for the number of the order of the index;
There is provided a tensor data calculation method comprising: a calculation processing step of performing a loop calculation on tensor data arranged on the data storage unit by the data arrangement processing step.

  According to the embodiment of the present invention, there is provided a technique for speeding up the process for factoring the sparse tensor and reducing the amount of memory and the amount of memory access communication required for the process.

It is a figure which shows the example of non-negative tensor factorization. It is a figure which shows the update process sequence of the factor matrix A in a dense tensor. It is a figure which shows the update process sequence of the factor matrix A in a missing value sparse tensor. It is a figure which shows the update process sequence of the factor matrix A in 0 value sparse tensor. It is a block diagram of the tensor data calculation apparatus in an embodiment of the present invention. It is a figure which shows the image of the data structure of tensor data. It is a figure which shows the update process procedure of the factor matrix A of a dense tensor in the case of hold | maintaining tensor data with a tertiary array. It is a figure which shows the update process sequence of the factor matrix A of a missing value sparse tensor in the case of hold | maintaining tensor data with a 3rd-order arrangement | sequence. It is a figure which shows the update process sequence of the factor matrix A of 0 value sparse tensor in the case of holding | maintaining tensor data by a 3rd-order arrangement | sequence. It is a figure for demonstrating performing factor matrix expansion | deployment for every j, fixing i regarding the factor matrix A. FIG. It is a figure which shows the information required with respect to the user 1. It is a figure which shows the data hold | maintained for the update of the factor matrix A. 10 is a diagram for explaining an example of how to set axes in the factor matrix B and the factor matrix C. FIG. It is a figure which shows the data hold | maintained for the update of the factor matrix B. FIG. It is a figure which shows the data for the update of each factor matrix. It is a figure for demonstrating the effect which prepares the sparse matrix for every axis | shaft. It is a figure which shows the expansion to a high-order tensor. Is a diagram illustrating the data held to update the factor matrix Z _k.

  Embodiments of the present invention will be described below with reference to the drawings. The embodiment described below is only an example, and the embodiment to which the present invention is applied is not limited to the following embodiment.

(Device configuration)
FIG. 5 shows a configuration of the tensor data calculation apparatus 10 according to the embodiment of the present invention. The tensor data calculation device 10 receives tensor data, performs non-negative tensor factorization, and outputs the resulting factor matrix data. As illustrated in FIG. 5, the tensor data calculation apparatus 10 includes a data input unit 11, a data arrangement processing unit 12, a data storage unit 13, a tensor factorization processing unit 14, and a data output unit 15.

  The data input unit 11 receives the tensor data and passes the tensor data to the data arrangement processing unit 12. The data arrangement processing unit 12 converts the tensor data into a sparse matrix and stores (arranges) the tensor data in the data storage unit 13 (memory or the like) with a data structure described in detail later. The tensor factorization processing unit 14 performs non-negative tensor factorization processing on the tensor data having the above data structure arranged in the data storage unit 13, and outputs the data of the factor matrix as the processing result to the data output unit 15. The data output unit 15 outputs the data.

  The tensor data calculation apparatus 10 according to the present embodiment can be realized by causing a computer to execute a program that describes the processing content described in the present embodiment. That is, the function of the tensor data calculation apparatus 10 is to execute a program corresponding to the processing executed by the tensor data calculation apparatus 10 using hardware resources such as a CPU, memory, and hard disk built in the computer. Can be realized. Further, the program can be recorded on a computer-readable recording medium (portable memory or the like), stored, or distributed. It is also possible to provide the program through a network such as the Internet or electronic mail.

  Hereinafter, the process in the tensor data calculation apparatus 10 will be described in more detail. In the following, the processing executed by the data arrangement processing unit 12 will be described in detail. In the following, first, a case where the cubic tensor data as shown in FIG. 1 is decomposed into three factor matrices A, B, and C will be described as an example. Thereafter, an example of extending to the Nth order (N is an integer of 3 or more) will be described.

(Tensor data structure)
First, a general tensor data structure handled by the tensor data calculation apparatus 10 will be described. In the tensor data calculation device 10, higher-order tensor data is stored and handled as an array for program processing. For example, the third order tensor data X _ijk is held in a third order array of X [I] [J] [K]. The image of the data structure in this case is as shown in FIG.

  In the data structure, in the normal dense matrix update process for factorization, the third-order tensor data is held in the third-order array of X [I] [J] [K] as described above, and FIG. As shown in FIG. 4, processing by a quadruple loop is performed. However, as described above, the dense tensor processing has a problem that the amount of calculation increases and the required memory area also increases.

  In this embodiment, sparse tensor data is targeted. In this case, as described above, elements that do not require calculation are omitted in the factorization update formula, and only the elements that require calculation are processed at high speed. Enable calculation.

The update process procedure of the factor matrix A in this case is as shown in FIGS. The update process is executed by the tensor factorization processing unit 14 of the tensor data calculation apparatus 10. As shown in FIG. 8, when the missing values sparse data, for example, in the updating of _{a ir, a ir} loops through the IxR against itself, j, with respect to the combination of k, j for i, k of the number of logs L Perform a calculation that turns the processing loop of _i . FIG. 9 shows the update processing procedure of the factor matrix A in the case of 0-value sparse data. As already described with reference to FIG. 4, in the case of zero-valued sparse data, loop calculation is separated and executed in accordance with the index dependency in the update formula for estimating the optimum factorization value. Specifically, the loop calculation related to the denominator calculation process and the loop calculation related to the numerator calculation process are executed separately. The processing with the most calculation amount is the calculation of the value of the molecule, and the calculation of the molecule is performed in the same manner as the missing value sparse tensor. That, in the calculation processing of the molecule, for example, in the updating of _{a ir,} loops through the IxR against _{a ir} itself, j, with respect to the combination of k, j for i, the processing loop of the log number _{L i} of k Perform a turn calculation. Regarding the denominator, as shown in FIG. 9, J times of loop processing and K times of loop processing are performed R times. As described above, the calculation amount of the denominator is negligible with respect to the total calculation amount.

At this time, it is necessary to hold the combination information of j, k, and l in each i. 8 and 9, the information is arranged in the data storage unit 13 as index data (index_j, index_k, index_l) as indicated by A. Further, X _ijk and ^ X _ijk are stored in the data storage unit 13 as vectors or similar indices. Similar data is arranged in the data storage unit 13 for the factor matrices B and C.

  In the present embodiment, the tensor factor decomposition processing unit 14 arranges the data of these indexes in the data storage unit 13 as a data structure that enables the update processing to be executed efficiently by the data arrangement processing unit 12. It is possible to perform matrix update of the executed sparse tensor at high speed.

  Hereinafter, in the present embodiment, data arranged in the data storage unit 13 will be described in more detail. The processing described below is common to the case of the missing value sparse tensor and the case of the zero value sparse tensor. However, the processing described below relates to the denominator and numerator calculation processing in the case of the missing value sparse tensor, and relates to the numerator calculation processing in the case of the zero value sparse tensor.

(Sparse matrix of tensor data: explanation using factor matrix A as an example)
As shown in the update procedure of FIGS. 8 and 9, the factor matrix A is updated with elements for each i. In the update loop calculation, i is the highest index.

  In this embodiment, tensor data is sparsed by fixing i and performing matrix expansion for each j. That is, the data arrangement processing unit 12 of the tensor data calculation apparatus 10 expands the tensor data as shown in FIG. 10A into a matrix for each j with i fixed as shown in FIG. 10B. To do. In the present embodiment, this is called sparse matrixing of tensor data. The point here is not to expand every j, but to fix i.

As shown in FIG. 10B, in the expanded result, when i = 1,
i = 1, [6, 0, 2, 4, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 2]
The vector Since the sparse tensor does not require processing of the 0 element, the data arrangement processing unit 12 arranges only the information of the necessary elements in the data storage unit 13. This information is shown as X _{ijk in} FIG.

At this time, since information on j and k is necessary, the information is held separately. This information is shown as (j, k) in FIG. For example, it is shown that the log data value 6 of the user 1 corresponds to (j, k) = (1, 1). In addition, FIG. 11 shows an example of the value of ^ _Xijk as information necessary for the update process. The value is a value that is sequentially updated as the factor matrix is updated.

  Note that k is the lowest index of the calculation loop, and j is the highest previous index.

As described above, since the factor matrix A is factorization with respect to I and the update process is performed with respect to i, the data arrangement processing unit 12 regarding the factor matrix A has the number of logs for each i and (j, k) for each i. , X _ijk value and ^ X _ijk value are arranged in the data storage unit 13.

Finally, in order to update the factor matrix A, data shown in FIGS. (X) _{ijk in} (c) is blank, which means that it is prepared in advance as a memory area for storing calculation results. Here, the sparse matrix data ((b) in FIG. 12) storing the combination of (j, k) is called an index sparse matrix.

  For the factor matrix A, the index sparse matrix as described above is held, and the update process is performed according to the procedure shown in FIGS. Thus, by replacing the sparse tensor with the sparse matrix representation and performing the processing, it is possible to increase the processing speed and reduce the memory usage.

When performing the data arrangement as described above, the data arrangement processing unit 12 sets each index in the order from the lowest index (example: k) to the previous index (example: j) in the calculation loop. The elements of the tensor data are ordered so that they are in the loop direction of the index, and according to the ordering, for the non-empty element in the tensor data, the value of the element (eg, X _ijk ) and the tensor data of the element An index value (for example, a value of (j, k)) from the lowest index indicating the upper position to the previous highest index is arranged on the data storage unit 13 and the non-empty element The process of calculating the count number (L _i ) and placing it in the data storage unit 13 is performed for each order of the highest index in the calculation loop (eg, the order of i = 1, 2, 3, 4). Execute only for the order number of the index (example: 4 for i = 1, 2, 3, 4).

In addition, the above-mentioned “order the loops in the order of each index in the order of each index (example: j) from the lowest index (example: k) of the calculation loop to the immediately preceding index”. For example, for (j, k) in each row i in FIGS. 10 (b) and 12 (b) (6 = 1 because j = 1 and k = 1 to L ₁ = 6 for i = 1). (When i = 2, j = 1, k = 2 to L ₂ = 4, so four are arranged).

By performing the calculation loop for updating by performing the data arrangement as described above, it is only necessary to refer to the index and element value by the count number (L _i ) of the element, and it is necessary to refer to an empty element. Therefore, the calculation process can be made efficient and the memory can be used efficiently. In addition, when accessing the value of a non-empty element, it is possible to access on the memory in order from the lowest to highest in the index loop, eliminating random access and increasing memory access speed. There is an effect that can be done.

(Tensor data sparse matrix: Factor matrices B and C)
As described above, the factor matrix A has sparse matrix rows in the I direction. The factor matrix B is processed by changing the axis as shown in FIGS. 13A to 13B in order to make the J direction a sparse matrix row. Similarly, the factor matrix C is processed by changing the axis as shown in FIGS. 13B to 13C in order to make the K direction a sparse matrix row.

FIG. 14 shows data held for the factor matrix B. As shown in FIG. 14, for the factor matrix B, an index sparse matrix of L _j , X _ijk , (k, i) is held, and a memory area of ^ X _ijk is secured. Similarly, for the factor matrix C, an index sparse matrix of L _k , X _ijk , (i, j) is retained, and a memory area of ^ X _ijk is secured. FIG. 15 collectively shows data for updating the factor matrices A, B, and C.

(Tense data sparse matrix: explanation of effects)
For example, if update processing of the factor matrix B based on the J-axis element is performed on X _ijk used in the factor matrix A, random access is made as shown in FIG.

  On the other hand, when a sparse matrix is prepared for each axis corresponding to a factor matrix as in the technique of the present embodiment, when updating the factor matrix, as shown in FIG. No access occurs, the communication cost is reduced, and the calculation process can be speeded up.

  That is, as in the present embodiment, by holding a sparse matrix data structure for update for each axis (mode), it is possible to perform processing for each mode by simple loop processing. In addition to simple loop processing, it is expected to increase the speed by suppressing random access of the memory, for example, improving the cache hit rate particularly in processing with large-scale data. The effect of suppressing random access is very effective, for example, in the case of parallel distributed processing such as GPU, it is only necessary to suppress the reference memory amount to the reference necessary amount in each thread in advance.

(Extension to Nth order)
The factorization example has been described so far by taking the third-order tensor as an example. However, the technique according to the present embodiment is not limited to the third-order tensor, and the N-order (N is 3 or more), which is an order larger than the third-order. (Integer) tensor.

In order N tensor data, it has a log number _{L, X i1 ... iN} tensor data, which, as shown in FIG. 17, consider each decomposed case to factor matrix _Z 1 ... _{Z N.} Incidentally, subscript X is represents a convenience "i1 ... iN" notation in the specification, which is intended _{"i 1} ... i _N".

The N-order case is the same as the case described for the third-order example, and the data arrangement processing unit 12 creates a sparse matrix expanded with respect to the corresponding axis for each of the factor matrices Z ₁ ... Z _N. .

Figure 18 (a) ~ (c) , shows the data for updating the factor matrix _{Z k.} As shown in FIGS. 18A to 18C, when the factor matrix of Z _k is updated, the number of logs related to _k is L _k and the tensor data of X _{i1... IN} is also held in the structure shown in FIG. Keep it. As for the index sparse matrix, as shown in (b), combinations of elements other than _k (i _{k + 1} ,..., I _N , i ₁ ,..., I _k−1 ) are held. This corresponds to the “each index from the lowest index of the calculation loop to the previous index of the highest one” described above. Also, a memory area of ^ X _{i1... IN} is secured. Thereby, it can be extended to a tensor factorization of arbitrary order N. That is, with the technique of the present embodiment, the index sparse matrix has a plurality of elements, and can be easily extended to a higher-order mode.

  The present invention is not limited to the above-described embodiments, and various modifications and applications are possible within the scope of the claims.

DESCRIPTION OF SYMBOLS 10 Tensor data calculation apparatus 11 Data input part 12 Data arrangement | positioning process part 13 Data storage part 14 Tensor factorization process part 15 Data output part

Claims (7)

A tensor data calculation device that performs a loop calculation on a plurality of indexes of tensor data for N-dimensional (N is an integer of 3 or more) tensor data,
The tensor data elements are ordered in the order of the loop direction of each index in the order of each index from the lowest index of the calculation loop to the previous highest index, and in the tensor data according to the ordering. For a non-empty element, the value of the element and the index value of the index from the lowest index indicating the position on the tensor data of the element to the immediately preceding index are arranged on the data storage unit, and Data placement processing means for calculating the number of non-empty elements and placing the data storage unit in the data storage unit, for each order of the highest index in the calculation loop, the number of the order of the index,
A tensor data calculation device comprising: calculation processing means for performing loop calculation on tensor data arranged on the data storage unit by the data arrangement processing means. The tensor data calculation apparatus according to claim 1, wherein the calculation processing unit performs factorization of the tensor data by loop calculation on the tensor data. The tensor data calculation according to claim 2, wherein the calculation processing unit performs loop calculation separately according to an index dependency in an update expression for estimating an optimum value of the factorization. apparatus. A tensor data calculation method executed by a tensor data calculation device that performs loop calculation on a plurality of indexes of tensor data for N-dimensional (N is an integer of 3 or more) tensor data,
The tensor data elements are ordered in the order of the loop direction of each index in the order of each index from the lowest index of the calculation loop to the previous highest index, and in the tensor data according to the ordering. For a non-empty element, the value of the element and the index value of the index from the lowest index indicating the position on the tensor data of the element to the immediately preceding index are arranged on the data storage unit, and A data placement processing step for calculating the number of non-empty elements and placing the data storage unit in the data storage unit for each order of the highest index in the calculation loop for the number of the order of the index;
A tensor data calculation method comprising: a calculation processing step of performing loop calculation on tensor data arranged on the data storage unit by the data arrangement processing step. 5. The tensor data calculation method according to claim 4, wherein, in the calculation processing step, the tensor data calculation device performs factorization of the tensor data by loop calculation on the tensor data. In the calculation processing step, the tensor data calculation device performs a loop calculation separately according to an index dependency in an update expression for estimating an optimum value of the factorization. 5. The tensor data calculation method according to 5.   The program for functioning a computer as each means in the tensor data calculation apparatus of any one of Claims 1 thru | or 3.

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