Computation of $M$-QDR decomposition of tensors and applications

TL;DR

This paper introduces a novel $M$-$\mathcal{QDR}$ tensor decomposition based on $M$-product, along with algorithms for its computation, and demonstrates its application in image compression.

Contribution

It proposes a new $M$-$\mathcal{QDR}$ decomposition for third-order tensors, with algorithms and applications in image compression.

Findings

Effective algorithms for $M$-$\mathcal{QDR}$ decomposition and tensor inverses.

Exact symbolic expressions for tensor outer inverses.

Successful application to lossy color image compression.

Abstract

The theory and computation of tensors with different tensor products play increasingly important roles in scientific computing and machine learning. Different products aim to preserve different algebraic properties from the matrix algebra, and the choice of tensor product determines the algorithms that can be directly applied. This study introduced a novel full-rank decomposition and $M$-$\mc{QDR}$ decomposition for third-order tensors based on $M$-product. Then, we designed algorithms for computing these two decompositions along with the Moore-Penrose inverse, and outer inverse of the tensors. In support of these theoretical results, a few numerical examples were discussed. In addition, we derive exact expressions for the outer inverses of tensors using symbolic tensor (tensors with polynomial entries) computation. We designed efficient algorithms to compute the Moore-Penrose inverse of symbolic tensors. The prowess of the proposed $M$-$\mc{QDR}$ decomposition for third-order tensors is applied to compress lossy color images.