CS decomposition and GSVD for tensors based on the T-product

TL;DR

This paper introduces tensor CS decomposition and GSVD based on the T-product, providing algorithms and demonstrating their effectiveness in image restoration tasks.

Contribution

It develops the first tensor CS decomposition and GSVD using the T-product, extending matrix decompositions to tensor analysis.

Findings

T-GSVD effectively solves tensor Tikhonov regularization

Algorithms for T-CSD and T-GSVD are proposed and analyzed

Numerical examples confirm the methods' effectiveness in image restoration

Abstract

This paper derives the CS decomposition for orthogonal tensors (T-CSD) and the generalized singular value decomposition for two tensors (T-GSVD) via the T-product. The structures of the two decompositions are analyzed in detail and are consistent with those for matrix cases. Then the corresponding algorithms are proposed respectively. Finally, T-GSVD can be used to give the explicit expression for the solution of tensor Tikhonov regularization. Numerical examples demonstrate the effectiveness of T-GSVD in solving image restoration problems.