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 the tensor product determines the algorithms that can be applied directly. This study introduced a novel full-rank decomposition and M-QDR decomposition for third-order tensors based on the M-product. Then we designed algorithms for computing these two decompositions along with the Moore-Penrose inverse and outer inverse of the tensors. Numerical examples are discussed in support of these theoretical results. In addition, we derive exact expressions for the outer inverses of tensors using a 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-QDR decomposition for third-order tensors is applied to compress lossy color images.
