JP2018097497A - Analysis apparatus, analysis method, and program - Google Patents

Abstract

PROBLEM TO BE SOLVED: To improve accuracy of analysis by performing non-negative value tensor complementation using adjacency relations between feature quantities.SOLUTION: An analysis apparatus includes: a data input unit for accepting an N-th level tensor X, a mask tensor M representing whether each element in the tensor X is a missing value or not, and an adjacency graph matrix Wrepresenting adjacency relations between feature amounts for each mode; a parameter updating unit for repeatedly updating a factor matrix Ato minimize the sum of a value of a cost function using generalized KL divergence between an estimated value X^ of the tensor X and the tensor X on the basis of the tensor X, the mask tensor M, and the adjacency graph matrix Wand a value of a penalty term concerning an error between the feature quantities in the factor matrix Ausing the adjacency graph matrix W; and a parameter output unit for outputting the factor matrix Afor each of modes n updated by the parameter updating unit.SELECTED DRAWING: Figure 1

Description

  The present invention belongs to the field of machine learning and data mining analysis, and particularly relates to a tensor decomposition technique.

  Non-negative tensor complementation (NTC) is a technique for estimating and restoring missing values included in data by tensor decomposition of observed values (Non-patent Documents 1 and 2). NTC uses non-negative tensor factorization (NTF) for tensor decomposition (Non-Patent Document 3, Non-Patent Document 4). NTF is a technique for learning a co-occurrence pattern consisting of a small number of non-negative values from data using the fact that the value of data is non-negative. Non-patent document 7 proposes NTC using generalized KL divergence (Non-patent document 5 and Non-patent document 6).

  Non-Patent Document 7 treats traffic flow data as a third-floor tensor consisting of attributes of observation position, time, and date, and performs analysis by NTC using generalized KL divergence, thereby congesting specific observation points. It is disclosed that the co-occurrence pattern of time and date to be extracted is extracted and the accuracy of missing value estimation is improved.

Y. Xu and W. Yin.A block coordinate descent method for regularized multiconvex optimization with applications to nonnegative tensor factorization and completion.SIAM Journal on Imaging Sciences, 6 (3): 1758-1789, 2013. T. G. Kolda and B. W. Bader.Tensor decompositions and applications.SIAM Review, 51 (3): 455-500, 2009. N. Gillis.The why and how of nonnegative matrix factorization.arXiv preprint arXiv: 1401.5226, 2014. A. Shashua and T. Hazan.Non-negative tensor factorization with applications to statistics and computer vision.In Proc.ICML, 2005. D. D. Lee and H. S. Seung. Algorithms for non-negative matrix factorization. In NIPS, 2000. A. Cichocki, R. Zdunek, A. H. Phan, and S. Amari.Nonnegative matrix and tensor factoriza- tions: applications to exploratory multi-way data analysis and blind source separation.Wiley, 2009. Takashi Takeuchi, Tadashi Naya, Nobuyoshi Ueda "Non-negative tensor interpolation using generalized KL divergence and its application to urban traffic data", 8th Forum on Data Engineering and Information Management Cai, D., He, X., Han, J., and Huang, T. S. Graph regularized nonnegative matrix factorization for data representation.IEEE Transactions on Pattern Analysis and Machine Intelligence 33, 8 (2011), 1548-1560.

  However, since the conventional NTC does not consider the relevance of the feature amounts included in the data attributes, there is a problem that the accuracy of the analysis is lowered when the observation values in the data are small. The relationship between feature amounts is information such as the distance between observation positions, the elapsed time between times, and the number of days elapsed between dates.

  The present invention has been made in view of the above points, and an object of the present invention is to improve the accuracy of analysis by performing non-negative tensor complementation using an adjacent relationship between feature quantities.

According to the disclosed technique, the N-th order tensor X composed of non-negative elements, the mask tensor M indicating whether each element in the tensor X is a missing value or an observed value, and the feature amount of each mode A data input unit that accepts an adjacency graph matrix W ⁽ⁿ⁾ representing the adjacency relationship;
The tensor X accepted by the data input unit, the mask tensor M, and the based on the adjacency graph matrix W ^(n), generalized KL divergence between the estimated value X ^ and the tensor X of the tensor X To minimize the sum of the value of the cost function using and the penalty term value for the error between the features in the factor matrix A ⁽ⁿ⁾ using the adjacency graph matrix W ⁽ⁿ⁾ A parameter updating unit that repeats updating the factor matrix A ⁽ⁿ⁾ for each;
A parameter output unit that outputs the factor matrix A ⁽ⁿ⁾ for each of the modes n updated by the parameter update unit;
Is provided.

  According to the disclosed technique, it is possible to improve the accuracy of analysis by performing non-negative tensor complementation using the adjacent relationship between feature quantities.

It is a block diagram which shows the structure of the analyzer which concerns on embodiment of this invention. It is a figure which shows the algorithm of the process in the analyzer which concerns on embodiment of this invention. It is a flowchart of the process which the analyzer which concerns on embodiment of this invention performs. It is a figure for demonstrating an effect.

  Hereinafter, an embodiment (this embodiment) of the present invention will be described with reference to the drawings. The embodiment described below is merely an example, and the embodiment to which the present invention is applied is not limited to the following embodiment.

<Outline of Embodiment of the Present Invention>
In this embodiment, the adjacency relationship between feature amounts is used, and the regularization technique described in Non-Patent Document 8 is applied to NTC, and the relationship between feature amounts (eg, distance between observation positions and time intervals). NTC that takes into account information such as elapsed time and days elapsed between dates). By using the adjacency relationship between the feature quantities, the parameters corresponding to the feature quantities in the adjacency relation have similar estimated values, and the accuracy of estimating missing values is improved.

<Principle of Embodiment of the Present Invention>
First, the principle in the present embodiment will be described.

NTC using generalized KL divergence minimizes the following loss function using the input N-th order tensors X and M and the initial value of the factor matrix A ⁽ⁿ⁾ (n = 1, ..., N) In order to solve the problem, A ⁽ⁿ⁾ (n = 1,..., N) is repeatedly updated. The symbol indicating the tensor is underlined.

The number of modes is N, and the length (number of features) of the feature quantity of each mode is I ₁ , ..., _N N-order non-negative tensor of I _N ,

とし、テンソルのインデクスとその集合をi = (i₁、…、i_N)、テンソルXのnモードアンフォールドをX ⁽ⁿ⁾とする。テンソルXの欠損を示すマスクテンソルをMとし、マスクテンソルの要素m_iを、

And the index of tensor and its set are i = (i ₁ ,..., I _N ), and the n-mode unfold of tensor X is X ⁽ⁿ⁾ . The mask tensor that indicates a defect in the tensor X is M , and the mask tensor element _mi is

とする。

And

Next, an adjacency graph expressing the adjacency relationship of feature amounts for each mode is determined. The adjacency graph matrix for the n-th mode is W ⁽ⁿ⁾ ∈ R ₊ ^{In × In,} and the similarity between the i-th and j-th feature quantities is represented by an element w _{i, i ′} ⁽ⁿ⁾ ∈ R ₊ . That is, the greater the value of w _{i, i ′} ⁽ⁿ⁾ , the higher the similarity.

  The tensor interpolation model uses CP decomposition that approximates the observed value by a linear sum of K factors. The number of factors in non-negative tensor interpolation is K in all modes. A factor matrix with the kth factor vector corresponding to the nth mode and the factor vector in the column vector,

と定め、 因子行列の集合をA = {A⁽¹⁾、A⁽²⁾、…、A⁽ⁿ⁾ 、…、A^(N)}とする。テンソルXの推定値を、 因子ベクトルの線形和、

And the set of factor matrices is A = {A ⁽¹⁾ , A ⁽²⁾ , ..., A ⁽ⁿ⁾ , ..., A ^(N) }. The estimate of tensor X , the linear sum of factor vectors,

と定める。コスト関数fをXとX^の要素毎の近似誤差の線形和、すなわち、

It is determined. The cost function f is a linear sum of approximation errors for each element of X and X ^ , that is,

とする。以上から、CP分解を用いた非負テンソル補完は次の最小化問題、

And From the above, non-negative tensor interpolation using CP decomposition is the following minimization problem:

と定められる。

It is determined.

Allowing X and A ⁽ⁿ⁾ (n = 1,..., N) to take only non-negative values is the meaning of the “non-negative value” in the method name. In addition, attribute co-occurrence patterns are extracted as a factor matrix A ⁽ⁿ⁾ (n = 1,..., N) for K factors.

  Generalized KL (Cullback Liver) divergence for each element

と定められる。

It is determined.

In the present embodiment, a penal term Ω (A ⁽ⁿ⁾ ) relating to an error between feature quantities in the factor matrix A ⁽ⁿ⁾ is determined using an adjacency graph and generalized KL divergence as follows.

定義からw_i,i' ⁽ⁿ⁾が非ゼロの値を保つ場合には、k番目の因子ベクトルのi番目の要素とi'番目の要素が同一の値を取るときに、この誤差はゼロとなる。NTCのコスト関数fと罰則項Ω(A⁽ⁿ⁾)のモード毎の和を用いて、本実施の形態におけるコスト関数gを定める。

If w _{i, i '} ⁽ⁿ⁾ keeps a non-zero value from the definition, this error is zero when the i-th element and i'-th element of the k-th factor vector take the same value. It becomes. The cost function g in the present embodiment is determined using the sum of the NTC cost function f and the penalty term Ω (A ⁽ⁿ⁾ ) for each mode.

以上から、本実施の形態における技術はgを最小化するA^*を求める技術と定められる。

From the above, the technique in the present embodiment is defined as a technique for obtaining A ^* that minimizes g.

＜本発明の実施の形態に係る解析装置の構成＞
次に、本実施の形態に係る解析装置１００の構成について説明する。本実施の形態に係る解析装置１００は、ＣＰＵと、ＲＡＭと、後述する解析処理ルーチンを実行するためのプログラムや各種データを記憶したＲＯＭと、を含むコンピュータで構成することが出来る。図１に示すように、この解析装置１００は、機能的にはデータ入力部１０と、演算部２０とを備えている。

<Configuration of analyzer according to embodiment of the present invention>
Next, the configuration of analysis apparatus 100 according to the present embodiment will be described. The analysis apparatus 100 according to the present embodiment can be configured by a computer including a CPU, a RAM, and a ROM that stores a program for executing an analysis processing routine described later and various data. As shown in FIG. 1, the analysis apparatus 100 functionally includes a data input unit 10 and a calculation unit 20.

The data input unit 10 inputs data X in tensor format and missing value instruction data (mask tensor) M in tensor format. The length (number of features, number of elements) of each mode of X is I ₁ ,..., I _N, and only non-negative (0 or more) values are allowed for the elements of X. M is allowed only when the length of each mode is I ₁ , ..., I _N , and the elements of M are limited to 0 or 1 only. The X element with an M value of 0 is treated as a missing value, and the 1 element is treated as an observed value. Further, matrix format data W ⁽ⁿ⁾ (n = 1,..., N) is input. W ⁽ⁿ⁾ is a matrix of length I _n × I _n , and the element of W ⁽ⁿ⁾ allows only non-negative values (0 or more).

Further, the data input unit 10 inputs a set of factor matrices A = {A ⁽¹⁾ , A ⁽²⁾ ,..., A ^(N) } as initial values of parameters. The element of A ⁽ⁿ⁾ only allows non-negative (0 or more) values. In addition, the data input unit 10 inputs the number T of parameter update repetitions.

  The calculation unit 20 includes an unfold calculation unit 26, a parameter update unit 30, and a parameter output unit 38.

For each mode n, the unfold calculation unit 26 n-mode unfold X _{(n) obtained by} unfolding the tensor X with respect to the mode n, and n-mode unfold obtained by unfolding the mask tensor M with respect to the mode n. Calculate M _(n) .

  The parameter update unit 30 includes a factor matrix update unit 32 and an iterative determination unit 36.

The parameter update unit 30 receives the n-mode unfold X _{(n) of} the tensor X calculated by the unfold calculation unit 26 and the n-mode unfold M _{(n) of} the mask tensor M by the processing of each unit described above. Based on the adjacency graph matrix W ⁽ⁿ⁾ , the factor matrix A ⁽ⁿ⁾ for each of the N modes is calculated so as to minimize the sum of the value of the cost function f and the sum of the penalty terms for each mode. Repeat updating. The factor number K is determined in advance. The cost function f, the mode n have the same number of dimensions, wherein the number I _n of, and consisting of factor vector a _k corresponding to the k-th factor consisting of non-negative elements ⁽ⁿ⁾ I _n × K factor matrices A the elements of the factor vector a _k ⁽ⁿ⁾ of ^(n), the product obtained by multiplying for each mode n, the value obtained by adding for each of the K factor, generalized representing the distance between the elements of the tensor X Expressed using the sum of KL divergence.

Here, a method for updating the factor matrix A ⁽ⁿ⁾ will be described in more detail.

In updating the factor matrix set A = {A ⁽¹⁾ , A ⁽²⁾ ,..., A ^(N) }, the same processing up to n = 1,.

A ⁽ⁿ⁾ is updated with the following calculation method. The n-mode unfold of the estimated value X ^ is

となる。 この時、

It becomes. At this time,

はクラーチ・ラオ積（Kahtri-Rao product）とし、

Is the Kahtri-Rao product,

とした。

It was.

とすると、 因子行列A⁽ⁿ⁾に関する最小化問題は、

Then the minimization problem for the factor matrix A ⁽ⁿ⁾ is

となる。
g(A⁽ⁿ⁾)は因子ベクトルa_k ⁽ⁿ⁾毎に分離可能であるから、因子ベクトルa_k ⁽ⁿ⁾毎に更新を行うことで、上記の最小化問題の最適解を求める。ここで、グラフラプラシアンを

It becomes.
Since g ^{(A (n))} can be separated for each factor vector a _k ^(n), by performing the update for each factor vector a _k ^(n), finding the optimal solution of the minimization problem. Where graph Laplacian

と定める。このとき対角行列D⁽ⁿ⁾は行と列の長さがI_nで対角成分がW⁽ⁿ⁾の行毎の和からなり、非対角成分は0の行列とする。すると因子ベクトルの最小解は以下の線形方程式を解くことで求められる。

It is determined. Diagonal in this case the diagonal matrix D ⁽ⁿ⁾ the length of the row and column I _n is the sum of each row of W ^(n), the non-diagonal elements to zero matrix. Then, the minimum solution of the factor vector can be obtained by solving the following linear equation.

この時、C⁽ⁿ⁾は次のように定めた。

At this time, C ⁽ⁿ⁾ was determined as follows.

上記の式において

In the above formula

は行列の要素毎の積と商である。つまり、上記の式のかっこ()内の式は、M _(n)とX _(n)との要素毎の積の演算結果と、推定値X^のnモードアンフォールドとの要素毎の商である。この更新則を用いて各因子行列A⁽ⁿ⁾を順番に更新する。

Is the product and quotient of each element of the matrix. In other words, the expression in parentheses () in the above expression is the quotient for each element of the result of the product of each element of M _(n) and X _(n) and the n-mode unfold of the estimated value X ^. is there. Each factor matrix A ⁽ⁿ⁾ is updated in order using this update rule.

In accordance with the method described above, the factor matrix update unit 32 first initializes the factor matrix A ⁽ⁿ⁾ for each of the modes n. Further, the factor matrix update unit 32, for each of the modes n of the tensor X , the n-mode unfold X _(n) and the n-mode unfold M _(n) calculated by the unfold calculation unit 26, and the adjacent graph matrix W ⁽ⁿ⁾ is used to minimize the sum of the cost function value and the penalty term value related to the error between the feature quantities in the factor matrix A ⁽ⁿ⁾ , according to the above equation (1). A ⁽ⁿ⁾ is updated. The value of the cost function, the factor matrix A ^(n), the product of elements of the Kurachi Rao product of mode n other than the mode n'factor matrix A ^(n'), n mode unfold X tensor X It is expressed using the sum of generalized KL divergence that represents the distance to the element of _(n) .

The iterative determination unit 36 causes each update of the factor matrix A ⁽ⁿ⁾ by the factor matrix update unit 32 to be repeated until a predetermined condition is satisfied. In the present embodiment, as a predetermined condition, the maximum parameter update repetition calculation count T is repeated.

The parameter output unit 38 outputs a set of factor matrices A = {A ⁽¹⁾ , A ⁽²⁾ ,..., A ^(N) } finally obtained by the operation of the parameter update unit 30.

  The analysis apparatus 100 according to the present embodiment can be realized, for example, by causing a computer to execute a program describing the processing content described in the present embodiment. That is, the functions of the analysis device 100 are realized by executing a program corresponding to the processing executed by the analysis device 100 using hardware resources such as a CPU, a memory, and a hard disk built in the computer. Is possible. 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.

<Operation of Analysis Device According to Embodiment of the Present Invention>
FIG. 2 shows a processing algorithm in the analysis apparatus 100 according to the present embodiment. The operation of the analysis apparatus 100 that performs processing according to the algorithm will be described along the flow (analysis processing routine) of FIG. FIG. 2 also shows step numbers corresponding to the flow of FIG.

When the data input unit 10 receives the tensor X , the mask tensor M , the number of factors K, the adjacency graph matrix W ⁽ⁿ⁾ , the initial value of each factor matrix A ⁽ⁿ⁾ , and the maximum parameter update iteration count T, the analysis apparatus 100 Executes the analysis processing routine shown in FIG. Note that the data received by the data input unit 10 is stored in a memory or the like of the analysis apparatus 100 as a computer and is appropriately read for processing.
First, in step S100, accepts the data input unit 10, based on the initial value of each factor matrix A ^(n), initializes each factor matrix A ^(n). Also, t is initialized to t = 1.

Next, in step S102, based on the tensor X and the mask tensor M received by the data input unit 10, the n mode unfold X _(n) of the tensor X and the n mode of the mask tensor M for each mode n. Calculate unfold M _(n) .

In step S104, the n-mode unfold X _{(n) of} the tensor X calculated in step S102, the n-mode unfold M _{(n) of} the mask tensor M , the factor matrix A ⁽ⁿ⁾ , and the adjacency graph matrix W Based on ⁽ⁿ⁾ , the factor matrix A ⁽ⁿ⁾ is updated based on the above equation (1) so as to minimize the sum of the value of the cost function and the value of the penalty term.

In step S106, it is determined whether or not the processing in step S104 has been performed for all factor matrices A ⁽ⁿ⁾ . If the t-th process has not been performed in step S104 factor matrix ^{A (n)} is present, the process returns to step S104, it performs the process of step S104 for the factor matrix A ^(n). On the other hand, when the t-th process of step S104 is performed for all factor matrices A ⁽ⁿ⁾ , the process proceeds to step S108.

  In step S108, it is determined whether the number of repetitions t = T. If t = T, the process proceeds to step S112. If t ≠ T, the process proceeds to step S110 and counts up to t = t + 1. Returning to step S104, the process is repeated.

In step S112, the factor matrix set A = {A ⁽¹⁾ , A ⁽²⁾ ,..., A ^(N) } finally updated in step S104 is output, and the process ends.

<Explanation of effects in experimental examples>
In the experimental example, the effectiveness of the proposed method is demonstrated using the Queson value estimation experiment (City pulse (http://www.ict-citypulse.eu/)) of traffic flow data measured at 419 points in Aachen, Denmark. confirmed.

The observation period of data is 61 days from August 1 to September 31, 2014. During this period, there are no holidays in the Danish calendar. Since the data is observed every 30 minutes, there are a maximum of 2,928 observations for each sensor if the data is complete. The observation value of the data was examined for each sensor, and it was determined as missing if 24-hour data was not completely observed. From the traffic flow data, we created a third-floor tensor X with the observation point (441 points), time (24 hours every 30 minutes), and date (61 days) as modes.

If the distance between the two observation points is less than 100 meters and the angle difference from the observation start point to the end point is 45 degrees or less, the adjacency matrix of the observation point mode is 1, so that w _{i, i '} ⁽¹⁾ is 1. Otherwise it was set to 0. In the adjacency matrix of the time mode, w _{i, i ′} ⁽²⁾ is 1 if two times are adjacent, and 0 otherwise. The adjacency matrix of the date mode is set to 1 if w _{i, i ′} ⁽³⁾ is 1 if two dates are adjacent, and 0 otherwise.

  In order to confirm the change in the accuracy of the missing value estimation with respect to the missing data rate, the observed values were randomly missing in addition to the actual missing values in the traffic flow data. Defect ratios of 10%, 50%, 70% and 90% were used. Five trials were performed for each setting, and observation values were randomly lost each time in each trial. 90% of the remaining observation values were used as training data and 10% were used as test data. The number of factors was 5, 10, and 20 and was determined by 5-fold cross validation.

  The results are shown in FIG. Prop. In the table of FIG. 4 is the result of the proposed method. Comparison methods include non-negative tensor completion using generalized KL divergence (NTC (gKL)), non-negative tensor completion using Euclidean distance (NTC (Eud)), and generalization for matrices in which tensors are unfolded with respect to sensor mode. Non-negative matrix interpolation (NMC (gKL)) using KL divergence, non-negative tensor factorization with gKL (NTF) treating missing values as 0, and average value of traffic flow data as estimated values The thing (Mean) was adopted.

  When the defect rate was 10%, the proposed method and NMC showed the best generalization performance in all indicators compared with other methods. On the other hand, when the defect rate was 30, 50, 70, and 90%, the generalization performance of NMC deteriorated, and the generalization performance of the proposed method became the best. From the above, the effect of the proposed method was confirmed.

(Summary of embodiment)
As described above, according to the present embodiment, the N-order tensor X composed of non-negative elements, the mask tensor M representing whether each element in the tensor X is a missing value or an observed value, And a data input unit that receives an adjacency graph matrix W ⁽ⁿ⁾ that expresses the adjacency relationship of feature quantities for each mode, the tensor X received by the data input unit, the mask tensor M , and the adjacency graph matrix W based on the ^(n), and the value of the cost function using generalized KL divergence between the estimated value X ^ and the tensor X of the tensor X, factor matrix using the adjacency graph matrix W ⁽ⁿ⁾ a ^a parameter updating unit that repeats updating the factor matrix A ⁽ⁿ⁾ for each of the modes n so as to minimize the sum of the penalties term value regarding the error between the feature quantities in ⁽ⁿ⁾ ; Updated by parameter update unit A parameter output portion for outputting the factor matrix A ⁽ⁿ⁾ for each of the modes n, analyzer with is provided with.

The penalty term is, for example, a term composed of a sum of products of elements of the adjacency graph matrix W ⁽ⁿ⁾ and generalized KL divergence between elements of the factor vector of the factor matrix A ⁽ⁿ⁾ .

The parameter updating unit performs updating for each factor vector of the factor matrix A ⁽ⁿ⁾ , for example, using an equation using a graph Laplacian of the adjacent graph matrix W ⁽ⁿ⁾ .

  Although the present embodiment has been described above, the present invention is not limited to the specific embodiment, and various modifications and changes can be made within the scope of the gist of the present invention described in the claims. Is possible.

DESCRIPTION OF SYMBOLS 10 Data input part 20 Operation part 26 Unfold calculation part 30 Parameter update part 32 Factor matrix update part 36 Repetition determination part 38 Parameter output part 100 Analysis apparatus

Claims (7)

N-th order tensor X consisting of non-negative elements, mask tensor M indicating whether each element in the tensor X is a missing value or an observed value, and an adjacency graph representing the adjacency relationship of features for each mode A data input unit that accepts a matrix W ⁽ⁿ⁾ ;
The tensor X accepted by the data input unit, the mask tensor M, and the based on the adjacency graph matrix W ^(n), generalized KL divergence between the estimated value X ^ and the tensor X of the tensor X To minimize the sum of the value of the cost function using and the penalty term value for the error between the features in the factor matrix A ⁽ⁿ⁾ using the adjacency graph matrix W ⁽ⁿ⁾ A parameter updating unit that repeats updating the factor matrix A ⁽ⁿ⁾ for each;
A parameter output unit that outputs the factor matrix A ⁽ⁿ⁾ for each of the modes n updated by the parameter update unit;
Analyzing device. The said penalty term is a term which consists of the sum of the product of the element of the said adjacency graph matrix W ⁽ⁿ⁾ and the generalized KL divergence between the elements of the factor vector of the said factor matrix A ^(n). Analysis device. The analysis apparatus according to claim 1, wherein the parameter update unit performs update for each factor vector of the factor matrix A ⁽ⁿ⁾ by an expression using a graph Laplacian of the adjacent graph matrix W ⁽ⁿ⁾ . An analysis method in an analysis apparatus including a data input unit, a parameter update unit, and a parameter output unit,
The data input unit is an N-th order tensor X composed of non-negative elements, a mask tensor M indicating whether each element in the tensor X is a missing value or an observed value, and adjacent feature quantities for each mode. Accepts an adjacency graph matrix W ⁽ⁿ⁾ representing the relationship,
The parameter updating unit, wherein said tensor X accepted by the data input unit, the mask tensor M, and based on the neighborhood graph matrix W ^(n), and the estimated values X ^ and the tensor X of the tensor X Minimize the sum of the cost function value using the generalized KL divergence between and the penalty term value for the error between feature quantities in the factor matrix A ⁽ⁿ⁾ using the adjacency graph matrix W ⁽ⁿ⁾ Repeating updating the factor matrix A ⁽ⁿ⁾ for each of the modes n,
An analysis method in which the parameter output unit outputs the factor matrix A ⁽ⁿ⁾ for each of the modes n updated by the parameter update unit. The said penalty term is a term which consists of the sum of the product of the element of the said adjacent graph matrix W ⁽ⁿ⁾ and the generalized KL divergence between the elements of the factor vector of the said factor matrix A ^(n). Analysis method. The analysis method according to claim 4 or 5, wherein the parameter updating unit updates each factor vector of the factor matrix A ⁽ⁿ⁾ by an expression using a graph Laplacian of the adjacent graph matrix W ⁽ⁿ⁾ .   The program for functioning a computer as each part of the analyzer of any one of Claims 1 thru | or 3.

Images