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

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 {X_k}_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 {X_k}_k=0^∞, that converges to A^†, given an appropriate initial value. Numerical experiments are presented to demonstrate the accuracy of the proposed method.