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

arXiv: https://arxiv.org/abs/2605.01467 · HTML: https://mlsanity.com/paper/2605.01467/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 6 of its 33 references that are in the map.
- First posted on arXiv 2 May 2026.

## Builds on (most cited first, 6 of 6)

- [Tensor Decomposition for Signal Processing and Machine Learning](https://mlsanity.com/paper/1607.01668/) (2016)
- [Exact tensor completion using t-SVD](https://mlsanity.com/paper/1502.04689/) (2015)
- [Tensor Methods in Computer Vision and Deep Learning](https://mlsanity.com/paper/2107.03436/) (2021)
- [Tensor Methods and Recommender Systems](https://mlsanity.com/paper/1603.06038/) (2016)
- [Nonlinear Transform Induced Tensor Nuclear Norm for Tensor Completion](https://mlsanity.com/paper/2110.08774/) (2021)
- [Quaternion Tensor Completion with Sparseness for Color Video Recovery](https://mlsanity.com/paper/2212.08361/) (2022)

## Abstract (arXiv)

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.