# Euclidean-Norm-Induced Schatten-p Quasi-Norm Regularization for Low-Rank Tensor Completion and Tensor Robust Principal Component Analysis

arXiv: https://arxiv.org/abs/2012.03436 · HTML: https://mlsanity.com/paper/2012.03436/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Dec 2020.

## Built on by (most cited first, 4 of 4)

- [DBA: Efficient Transformer with Dynamic Bilinear Low-Rank Attention](https://mlsanity.com/paper/2211.16368/) (2022)
- [Spectral-Spatial Extraction through Layered Tensor Decomposition for Hyperspectral Anomaly Detection](https://mlsanity.com/paper/2503.05183/) (2025)
- [Low-Rank Tensor Recovery via Variational Schatten-p Quasi-Norm and Jacobian Regularization](https://mlsanity.com/paper/2506.22134/) (2025)
- [An Interdisciplinary and Cross-Task Review on Missing Data Imputation](https://mlsanity.com/paper/2511.01196/) (2025)

## Abstract (arXiv)

The nuclear norm and Schatten-p quasi-norm are popular rank proxies in low-rank matrix recovery. However, computing the nuclear norm or Schatten-p quasi-norm of a tensor is hard in both theory and practice, hindering their application to low-rank tensor completion (LRTC) and tensor robust principal component analysis (TRPCA). In this paper, we propose a new class of tensor rank regularizers based on the Euclidean norms of the CP component vectors of a tensor and show that these regularizers are monotonic transformations of tensor Schatten-p quasi-norm. This connection enables us to minimize the Schatten-p quasi-norm in LRTC and TRPCA implicitly via the component vectors. The method scales to big tensors and provides an arbitrarily sharper rank proxy for low-rank tensor recovery compared to the nuclear norm. On the other hand, we study the generalization abilities of LRTC with the Schatten-p quasi-norm regularizer and LRTC with the proposed regularizers. The theorems show that a relatively sharper regularizer leads to a tighter error bound, which is consistent with our numerical results. Particularly, we prove that for LRTC with Schatten-p quasi-norm regularizer on d-order tensors, p=1/d is always better than any p>1/d in terms of the generalization ability. We also provide a recovery error bound to verify the usefulness of small p in the Schatten-p quasi-norm for TRPCA. Numerical results on synthetic data and real data demonstrate the effectiveness of the regularization methods and theorems.