Cross Tensor Approximation Methods for Compression and Dimensionality Reduction

Salman Ahmadi-Asl, Cesar F. Caiafa, Andrzej Cichocki, Anh Huy Phan, Toshihisa Tanaka, Ivan Oseledets, Jun Wang

Research output: Contribution to journalArticlepeer-review


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 so that their structures, such as nonnegativity, smoothness, or sparsity, can be potentially preserved. This paper reviews and extends 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 Tucker and tubal SVD (t-SVD) models while we provide references to other decompositions such as Tensor Train (TT), Hierarchical Tucker (HT), and Canonical Polyadic (CP) decompositions. We evaluate the performance of the CTA algorithms by extensive computer simulations to compress color and medical images and compare their performance.

Original languageEnglish
Pages (from-to)150809-150838
Number of pages30
JournalIEEE Access
Publication statusPublished - 2021


  • CUR algorithms
  • cross approximation
  • randomization
  • tensor decomposition
  • tubal SVD


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