Tucker-3 decomposition with sparse core array using a penalty function based on Gini-index

• Published in: Japanese journal of statistics and data science Vol. 5; no. 2; pp. 675 - 700
• Main Authors: Tsuchida, Jun, Yadohisa, Hiroshi
• Format: Journal Article
• Language: English
• Published: Singapore Springer Nature Singapore 01.12.2022
• Subjects: Chemistry and Earth Sciences; Computer Science; Economics; Finance; Health Sciences; Humanities; Insurance; Law; Management; Mathematics and Statistics; Medicine; Original Paper; Physics; Statistical Theory and Methods; Statistics; Statistics and Computing/Statistics Programs; Statistics for Business; Statistics for Engineering; Statistics for Life Sciences; Statistics for Social Sciences; Tensor decomposition; Alternating least squares; Dimensional reduction; Regularized method
• ISSN: 2520-8756; 2520-8764
• DOI: 10.1007/s42081-022-00179-7

Tucker-3 decomposition is a dimension reduction method for tensor data, similar to principal component analysis. One of the characteristics of Tucker-3 is the core array, which represents the interactions between low-dimensional spaces. However, it is difficult to interpret the result when the number of elements in the core array is large. One solution to this problem is using sparse estimation, such as the L1 regularization method, for the core array. However, some regularization methods often sacrifice the model fit too much. To solve this issue, we propose a novel estimation method for Tucker-3 decomposition with a penalty function based on the Gini index, which is a measure of sparsity and variance. Maximizing the Gini index is expected to obtain an estimated value for the core array that is easy to interpret. Moreover, the model fitted to the data will not involve shrinkage much because the Gini index is a measure of variance, which is one of the model fit measures of Tucker-3. The nonconvex problem poses a challenge when using the proposed penalty function based on the Gini index. To address this problem, we develop a majorization–minimization algorithm. From a numerical example, we revealed that the performance (the precision and accurate prediction of zero cells) of our method is superior to that of the estimation method with existing penalties, such as the L1 penalty, smoothly clipped absolute deviation, and minimax concave penalty.