# A novel non-convex minimax p-th order concave penalty function approach to low-rank tensor completion

arXiv: https://arxiv.org/abs/2502.19979 · HTML: https://mlsanity.com/paper/2502.19979/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 37 references that are in the map.
- First posted on arXiv 27 Feb 2025.

## Builds on (most cited first, 8 of 8)

- [Tensor Robust Principal Component Analysis with A New Tensor Nuclear Norm](https://mlsanity.com/paper/1804.03728/) (2018)
- [Exact tensor completion using t-SVD](https://mlsanity.com/paper/1502.04689/) (2015)
- [Novel methods for multilinear data completion and de-noising based on tensor-SVD](https://mlsanity.com/paper/1407.1785/) (2014)
- [Framelet Representation of Tensor Nuclear Norm for Third-Order Tensor Completion](https://mlsanity.com/paper/1909.06982/) (2019)
- [A General Iterative Shrinkage and Thresholding Algorithm for Non-convex Regularized Optimization Problems](https://mlsanity.com/paper/1303.4434/) (2013)
- [Multi-dimensional imaging data recovery via minimizing the partial sum of tubal nuclear norm](https://mlsanity.com/paper/1712.05870/) (2017)
- [Tensor N-tubal rank and its convex relaxation for low-rank tensor recovery](https://mlsanity.com/paper/1812.00688/) (2018)
- [Tensor Recovery Based on A Novel Non-convex Function Minimax Logarithmic Concave Penalty Function](https://mlsanity.com/paper/2206.13506/) (2022)

## Abstract (arXiv)

The low-rank tensor completion (LRTC) problem aims to reconstruct a tensor from partial sample information, which has attracted significant interest in a wide range of practical applications such as image processing and computer vision. Among the various techniques employed for the LRTC problem, non-convex relaxation methods have been widely studied for their effectiveness in handling tensor singular values, which are crucial for accurate tensor recovery. While the minimax concave penalty (MCP) non-convex relaxation method has achieved promising results in tackling the LRTC problem and gained widely adopted, it exhibits a notable limitation: insufficient penalty on small singular values during the singular value handling process, resulting in inefficient tensor recovery. To address this issue and enhance recovery performance, a novel minimax p-th order concave penalty (MPCP) function is proposed. Based on this novel function, a tensor p-th order τ norm is proposed as a non-convex relaxation for tensor rank approximation, thereby establishing an MPCP-based LRTC model. Furthermore, theoretical convergence guarantees are rigorously established for the proposed method. Extensive numerical experiments conducted on multiple real datasets demonstrate that the proposed method outperforms the state-of-the-art methods in both visual quality and quantitative metrics.