Cross Tensor Approximation Methods for Compression and Dimensionality Reduction
Salman Ahmadi‐Asl, César F. Caiafa, Andrzej Cichocki, Anh Huy Phan, Toshihisa Tanaka, Ivan Oseledets, Jun Wang
Abstract
Cross Tensor Approximation (CTA) is a generalization of Cross/skeleton matrix and CUR Matrix Approximation (CMA) and is a suitable tool for fast low-rank tensor approximation. It facilitates interpreting the underlying data tensors and decomposing/compressing tensors in such a way that their structures such as nonnegativity, smoothness, or sparsity can be potentially preserved. In this paper, we review and extend state-of-the-art deterministic and randomized algorithms for CTA with intuitive graphical illustrations. We discuss several possible generalizations of the CMA to tensors, including CTAs: based on fiber selection, slice-tube selection, and lateral-horizontal slice selection. The main focus is on the CTA algorithms using the Tucker and t-SVD models while we provide references to other decompositions such as Tensor Train (TT), Hierarchical Tucker (HT) and Canonical Polyadic Decomposition (CPD). We evaluate the performance of CTA algorithms by extensive computer simulations to compress color and medical images and compare their performance.