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Low T-Phase Rank Approximation of Third Order Tensors

Taehyeong Kim, Hayoung Choi, Yimin Wei

TL;DR

This work develops a phase-centric framework for low-rank approximation of sectorial third-order tensors under the tensor $T$-product by introducing canonical T-phases and the T-phase rank. A key contribution is a phase-majorization inequality for the tensor geometric mean, enabled by block-circulant lifting, which yields an exact optimal value and a constructive half-phase truncation in the positive-imaginary regime. The authors extend classical phase inequalities to tensors, provide a practical half-phase truncation algorithm, and connect the theory to tensor MIMO LTI stability via a tensor small-phase theorem. The results offer a phase-sensitive alternative to magnitude-based $T$-SVD approaches, with implications for compression, denoising, and robust stability analysis in tensor-based multiway data and control 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.

Low T-Phase Rank Approximation of Third Order Tensors

TL;DR

This work develops a phase-centric framework for low-rank approximation of sectorial third-order tensors under the tensor -product by introducing canonical T-phases and the T-phase rank. A key contribution is a phase-majorization inequality for the tensor geometric mean, enabled by block-circulant lifting, which yields an exact optimal value and a constructive half-phase truncation in the positive-imaginary regime. The authors extend classical phase inequalities to tensors, provide a practical half-phase truncation algorithm, and connect the theory to tensor MIMO LTI stability via a tensor small-phase theorem. The results offer a phase-sensitive alternative to magnitude-based -SVD approaches, with implications for compression, denoising, and robust stability analysis in tensor-based multiway data and control systems.

Abstract

We study low T-phase-rank approximation of sectorial third-order tensors 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.
Paper Structure (5 sections, 19 theorems, 117 equations, 1 figure, 1 algorithm)

This paper contains 5 sections, 19 theorems, 117 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. T-phase gauges and T-phase inequalities
  4. Low T-phase-rank approximation via geometric means
  5. Phase of tensor MIMO LTI systems and small phase theorem

Key Result

Lemma 2.1

Let $\mathscr{A}\in\mathbb{C}^{m\times n\times p}$ and let $\bm F_p$ be the $p\times p$ unitary DFT matrix. Then where $\bm A_1,\ldots,\bm A_p$ are the frontal slices of the FFT of $\mathscr{A}$ along the third mode.

Figures (1)

  • Figure 1: Bode plot of the tensor MIMO system

Theorems & Definitions (39)

  • Lemma 2.1: Fourier block diagonalization; see kilmer2011factorization
  • Theorem 2.2: Sectorial tensor decomposition; ding2025tensor
  • Definition 2.3: Canonical T-phases
  • Proposition 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • proof
  • Theorem 3.4
  • ...and 29 more