Low T-Phase Rank Approximation of Third Order Tensors

TL;DR

This paper introduces a novel approach for low T-phase-rank approximation of third-order tensors using tensor T-product, canonical T-phases, and phase-majorization inequalities, with applications to MIMO systems.

Contribution

It defines canonical T-phases and T-phase rank, formulates the approximation problem, and derives tensor phase inequalities and an explicit optimal approximation formula.

Findings

Exact optimal-value formula in positive-imaginary regime

Explicit optimal half-phase truncation family

Tensor phase inequalities and small phase theorem for MIMO systems

Abstract

We study low T-phase-rank approximation of sectorial third-order tensors $\mathscr{A}\in\mathbb{C}^{n\times n\times p}$ under the tensor T-product. We introduce canonical T-phases and T-phase rank, and formulate the approximation task as minimizing a symmetric gauge of the canonical phase vector under a T-phase-rank constraint. Our main tool is a tensor phase-majorization inequality for the geometric mean, obtained by lifting the matrix inequality through the block-circulant representation. In the positive-imaginary regime, this yields an exact optimal-value formula and an explicit optimal half-phase truncation family. We further establish tensor counterparts of classical matrix phase inequalities and derive a tensor small phase theorem for MIMO linear time-invariant systems.