T-SVD Based Non-convex Tensor Completion and Robust Principal Component Analysis

TL;DR

This paper introduces a non-convex approach for tensor completion and robust PCA using T-SVD, which reduces bias and improves accuracy over traditional convex methods, supported by theoretical analysis and experiments.

Contribution

It proposes a novel non-convex tensor rank surrogate and sparsity measure, along with a majorization minimization algorithm for improved tensor recovery.

Findings

Outperforms existing methods on natural images

Effective on hyperspectral image data

Theoretically justified convergence and properties

Abstract

Tensor completion and robust principal component analysis have been widely used in machine learning while the key problem relies on the minimization of a tensor rank that is very challenging. A common way to tackle this difficulty is to approximate the tensor rank with the $\ell_1-$norm of singular values based on its Tensor Singular Value Decomposition (T-SVD). Besides, the sparsity of a tensor is also measured by its $\ell_1-$norm. However, the $\ell_1$ penalty is essentially biased and thus the result will deviate. In order to sidestep the bias, we propose a novel non-convex tensor rank surrogate function and a novel non-convex sparsity measure. In this new setting by using the concavity instead of the convexity, a majorization minimization algorithm is further designed for tensor completion and robust principal component analysis. Furthermore, we analyze its theoretical properties. Finally, the experiments on natural and hyperspectral images demonstrate the efficacy and efficiency of our proposed method.