Computation of $M$-QDR decomposition of tensors and applications
Krushnachandra Panigrahy, Biswarup Karmakar, Jajati Keshari Sahoo, Ratikanta Behera, Ram N. Mohapatra
TL;DR
This work tackles the challenge of tensor decomposition under the $M$-product to preserve algebraic structure in third-order tensors. It introduces two decompositions, FRD and $M$-$\mathcal{QDR}$, and develops algorithms for computing Moore-Penrose and outer inverses, including symbolic and multirank cases. The methods are validated through numerical examples and applied to lossy color image compression, demonstrating practical impact. Overall, the work provides a robust framework for exact and efficient tensor factorization, inverses, and symbolic computation with potential benefits for high-dimensional data processing.
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.