Quaternion tensor low rank Quaternion tensor low-rank approximation using a family of non-convex norms

TL;DR

This paper introduces novel non-convex norm-based methods for low-rank approximation of quaternion tensors, improving accuracy over traditional convex approaches in applications like inpainting and denoising.

Contribution

It proposes two new approaches using quasi-norms and non-convex norms for quaternion tensor low-rank approximation, extending existing tensor models.

Findings

Effective tensor approximation demonstrated in numerical experiments.

Outperforms convex nuclear norm methods in inpainting and denoising.

Theoretical guarantees support the proposed methods' efficiency.

Abstract

In this paper, we propose a new approaches for low rank approximation of quaternion tensors \cite{chen2019low,zhang1997quaternions,hamilton1866elements}. The first method uses quasi-norms to approximate the tensor by a low-rank tensor using the QT-product \cite{miao2023quaternion}, which generalizes the known L-product to N-mode quaternions. The second method involves Non-Convex norms to approximate the Tucker and TT-rank for the completion problem. We demonstrate that the proposed methods can effectively approximate the tensor compared to the convexifying of the rank, such as the nuclear norm. We provide theoretical results and numerical experiments to show the efficiency of the proposed methods in the Inpainting and Denoising applications.