Iterative Methods for Computing the T-Square Root of Third-Order Tensors

TL;DR

This paper introduces iterative algorithms for computing the principal square root of third-order tensors within the T-product framework, with applications in image processing and new tensor metrics.

Contribution

It proposes tensor extensions of Newton and Denman--Beavers iterations with convergence guarantees, and introduces tensor-based image processing techniques and a new tensor metric.

Findings

Algorithms exhibit rapid convergence in numerical experiments.

Tensor methods outperform classical techniques in structural preservation.

New tensor metric (Tensor Bures--Wasserstein) is mathematically validated.

Abstract

We develop and analyze iterative methods for computing the principal square root of third-order tensors under the T-product framework. Tensor extensions of the Newton iteration (quadratic convergence) and the Denman--Beavers iteration (geometric convergence with simultaneous computation of the inverse square root) are proposed, with rigorous convergence guarantees established via the Fourier-domain block-diagonalization of the T-product. We apply these methods to image processing, introducing Tensor Decorrelated Grayscale conversion, T-Whitening, and optimal color transfer under the T-product geometry. We also formulate the Tensor Bures--Wasserstein distance and prove it defines a valid metric on the space of T-positive definite tensors. Numerical experiments confirm rapid convergence and demonstrate that the proposed tensor-based techniques offer improved structural preservation and cross-channel decorrelation compared to classical methods.