Convergence analysis of iterative methods for computing the T-pseudoinverse of complete full-rank third-order tensors based on the T-product

Abstract

This paper proposes an iterative approach for estimating the T-pseudoinverse of a third-order tensor A. The T-pseudoinverse A† is defined as a generalization of the classical pseudoinverse for matrices. In this work, we present an efficient iterative method to estimate A† based on an iterative formula derived from Li and Li’s work on matrices. Additionally, we employ the T-product as the tensor multiplication operation. This iterative method avoids the tedious task of computing the T-pseudoinverse using singular value decomposition. Firstly, we demonstrate that if A is an invertible tensor, the proposed iterative method, represented by the sequence {Xk}k=0∞, converges to the inverse tensor of A, for a suitable initial value. Furthermore, for a complete full-rank tensor A, we propose a novel iterative method based on the sequence {Xk}k=0∞, that converges to A†, given an appropriate initial value. Numerical experiments are presented to demonstrate the accuracy of the proposed method.