Computation of generalized inverses of tensors via t‐product

Abstract

Generalized inverses of tensors play increasingly important roles in computational mathematics and numerical analysis. It is appropriate to develop the theory of generalized inverses of tensors within the algebraic structure of a ring. In this paper, we study different generalized inverses of tensors over a commutative ring and a noncommutative ring. Several numerical examples are provided in support of the theoretical results. We also propose algorithms for computing the inner inverses, the Moore–Penrose inverse, and weighted Moore–Penrose inverse of tensors over a noncommutative ring. The prowess of some of the results is demonstrated by applying these ideas to solve an image deblurring problem.

Tensor LU and QR decompositions and their randomized algorithms

“…A multiplication is presented for third-order tensors. Suppose that we have two tensors A ∈ R m×n×p and B ∈ R n×s×p and denote their frontal faces respectively as A (k) ∈ R m×n and B (k) ∈ R n×s , k = 1, 2, • • • , p. The operations bcirc, unfold and fold can be defined as [18,23,24], bcir c(A) :=      A (1) A (p) • • • A (2) A (2) A (1) • • • A (3) . .…”

“…In other words, every frontal slice of D is a weight sum of frontal slices of A, with the weights being some power of ω = e −2πı/n , ω is the primitive n-th root of unity in which ı = √ −1. Therefore, D (1) , D (2) , • • • , D (p) are lower-triangular (upper-triangular) matrices if and only if A (1) , A (2) , • • • , A (p) are lower-triangular (upper-triangular) matrices.…”

“…Definition 10 (Tensor LU decomposition) Let A ∈ C m×n×p be a complex tensor. Let D ∈ C m×n×p be the unique tensor determined by Lemma 1: bcir c(A) = (Fp ⊗ Im)diag(D (1) , D (2) (k) , k = 1, 2, . .…”

“…, p be the complete pivoting LU decomposition of corresponding frontal slices D (k) , where LD (k) ∈ R m×q , UD (k) ∈ R q×n , q = min(m, n). Then the tensors PA, QA, LA, UA defined by (1) , PD (2) (1) , QD (2) (1) , LD (2) (1) , UD (2)…”

“…Recently, the tensor T-product has been proved to be a useful tool in many real applications [1,2,7,8,6,5,13,21,18,23,24,28,29,30,31,32,33,34,37,46,48]. Wang et al [40] investigate the tensor neural network models based on the tensor singular value decomposition (T-SVD).…”