# Low-Rank Tensor Completion Based on Fractional Regularization with Ky Fan p-k Norm

arXiv: https://arxiv.org/abs/2606.19046 · HTML: https://mlsanity.com/paper/2606.19046/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 14 of its 54 references that are in the map.
- First posted on arXiv 17 Jun 2026.

## Builds on (most cited first, 14 of 14)

- [Tensor Ring Decomposition](https://mlsanity.com/paper/1606.05535/) (2016)
- [Tensor Robust Principal Component Analysis with A New Tensor Nuclear Norm](https://mlsanity.com/paper/1804.03728/) (2018)
- [Multi-dimensional imaging data recovery via minimizing the partial sum of tubal nuclear norm](https://mlsanity.com/paper/1712.05870/) (2017)
- [Exact Low Tubal Rank Tensor Recovery from Gaussian Measurements](https://mlsanity.com/paper/1806.02511/) (2018)
- [A Scale Invariant Approach for Sparse Signal Recovery](https://mlsanity.com/paper/1812.08852/) (2018)
- [Limited-angle CT reconstruction via the L1/L2 minimization](https://mlsanity.com/paper/2006.00601/) (2020)
- [Quaternion Nuclear Norm minus Frobenius Norm Minimization for color image reconstruction](https://mlsanity.com/paper/2409.07797/) (2024)
- [Multi-channel Nuclear Norm Minus Frobenius Norm Minimization for Color Image Denoising](https://mlsanity.com/paper/2209.08094/) (2022)
- [A class of null space conditions for sparse recovery via nonconvex, non-separable minimizations](https://mlsanity.com/paper/1710.07348/) (2017)
- [Analysis and algorithms for some compressed sensing models based on L1/L2 minimization](https://mlsanity.com/paper/2007.12821/) (2020)
- [Low-rank matrix recovery with Ky Fan 2-k-norm](https://mlsanity.com/paper/1904.05590/) (2019)
- [Finding the largest low-rank clusters with Ky Fan 2-k-norm and ℓ1-norm](https://mlsanity.com/paper/1403.5901/) (2014)
- [An equivalent reformulation and multi-proximity gradient algorithms for a class of nonsmooth fractional programming](https://mlsanity.com/paper/2311.00957/) (2023)
- [Quaternion Nuclear Norms Over Frobenius Norms Minimization for Robust Matrix Completion](https://mlsanity.com/paper/2504.21468/) (2025)

## Abstract (arXiv)

This paper addresses low-rank tensor completion (LRTC) by proposing a novel nonconvex surrogate, namely the ratio of the tensor nuclear norm to the tensor Ky Fan p-k norm (TNPK), to accurately approximate the tensor tubal rank. The TNPK possesses appealing properties, including scale invariance, parameter flexibility, and the existence of closed-form solutions under specific choices of p and k. With specific parameter settings of p and k, it reduces to the ratio of the tensor nuclear norm to the tensor Ky Fan k norm (TNK) or the ratio of the tensor nuclear norm to the tensor Frobenius norm (TNF). We construct a LRTC model and, under the tensor null space property (NSP), prove that low-rank tensors are local minimizers of the proposed model. Moreover, we derive the proximal operator of the Ky Fan p-k inverse-norm and further develop an efficient alternating direction method of multipliers (ADMM) algorithm with guaranteed subsequential convergence under mild conditions. Extensive experiments on synthetic and real-world datasets validate the superior performance of our method against state-of-the-art competitors.