Learnable Scaled Gradient Descent for Guaranteed Robust Tensor PCA

TL;DR

This paper introduces RTPCA-SGD, an efficient gradient descent method within the t-SVD framework for robust tensor PCA, providing theoretical guarantees and a learnable model for improved practical performance.

Contribution

It proposes a scalable, theoretically guaranteed gradient descent approach for tensor PCA using t-SVD, along with a learnable deep unfolding model for enhanced practical results.

Findings

RTPCA-SGD achieves linear convergence to the true low-rank tensor.

The method outperforms existing TNN-based approaches in efficiency.

Numerical experiments validate superior performance on synthetic and real datasets.

Abstract

Robust tensor principal component analysis (RTPCA) aims to separate the low-rank and sparse components from multi-dimensional data, making it an essential technique in the signal processing and computer vision fields. Recently emerging tensor singular value decomposition (t-SVD) has gained considerable attention for its ability to better capture the low-rank structure of tensors compared to traditional matrix SVD. However, existing methods often rely on the computationally expensive tensor nuclear norm (TNN), which limits their scalability for real-world tensors. To address this issue, we explore an efficient scaled gradient descent (SGD) approach within the t-SVD framework for the first time, and propose the RTPCA-SGD method. Theoretically, we rigorously establish the recovery guarantees of RTPCA-SGD under mild assumptions, demonstrating that with appropriate parameter selection, it achieves linear convergence to the true low-rank tensor at a constant rate, independent of the condition number. To enhance its practical applicability, we further propose a learnable self-supervised deep unfolding model, which enables effective parameter learning. Numerical experiments on both synthetic and real-world datasets demonstrate the superior performance of the proposed methods while maintaining competitive computational efficiency, especially consuming less time than RTPCA-TNN.