Quaternion Nonlinear Transform-Induced Nuclear Norm for Low-Rank Tensor Completion

TL;DR

This paper introduces a novel quaternion nonlinear transform-induced tensor nuclear norm for low-rank tensor completion, effectively capturing intrinsic correlations in quaternion-valued data like color images and videos.

Contribution

It extends nonlinear tensor nuclear norm models to the quaternion domain using a real embedding, enabling better modeling of inter-channel dependencies in color data.

Findings

Outperforms existing tensor completion methods on color video inpainting datasets.

Provides a tractable optimization framework with convergence guarantees.

Effectively models quaternion-valued data for improved low-rank tensor recovery.

Abstract

Tensor completion has emerged as a powerful framework for recovering missing data in multidimensional signals by exploiting low-rank tensor structures. Among existing approaches, linear transform-based tensor nuclear norm (TNN) methods have achieved considerable success by enforcing low-rankness on transformed frontal slices. However, the low-rank structure revealed by linear transforms remains inherently limited. To better capture intrinsic correlations, nonlinear transform-based TNN (NTTNN) models have been proposed, significantly enhancing low-rank representation through composite transforms. Despite their effectiveness, existing NTTNN methods are restricted to real-valued tensors and fail to model quaternion-valued data, which are essential for preserving inter-channel dependencies in color images and videos. Extending nonlinear TNN models to the quaternion domain is challenging due to the non-commutativity of quaternion multiplication and the complexity of quaternion singular value decomposition. To address the limitations encountered in prior works, we propose a quaternion nonlinear transform-induced tensor nuclear norm (QNTTNN) via a real embedding of quaternions, enabling tractable nuclear norm definitions and efficient optimization. Building upon QNTTNN, we formulate a quaternion tensor completion model and develop a proximal alternating minimization algorithm with rigorous convergence guarantees. Extensive experiments on benchmark color video inpainting datasets validate the superior performance of the proposed method over existing approaches.