Significant and Trivial Dependencies Separation in Data Tensor by the Projection Method

Abstract

The problem of data structure analysis through their multidimensional representation as a d-dimensional tensor is considered to assess dependencies on influencing factors in the decision-making process. The higher-order Singular Value Decomposition (SVD) is developed as a d-SVD schema to identify significant and trivial dependencies. The d-SVD includes the SVD of the tensor reshaped as a matrix and the SVDs of reduced size of the previous SVD vectors reshaped as matrices. The entropy of the distribution of the Singular Values (SVs) of the vectors’ decomposition is used for the separation of the significant and trivial vectors, in contrast to the commonly used approach based on the magnitude analysis of SVs. The singular projection in the significant vector space in selected dimensions gives the tensor’s low-rank approximation without loss of information in comparison with the truncated SVD. The tensor projection on a vector subspace of reduced dimension can be obtained by using a part of the SVs and the corresponding vectors as an alternative to the commonly used averaging. It was shown that data prediction in the subspace of the significant vectors allows stable assessments of the predicted values to be obtained.