Online Nonconvex Robust Tensor Principal Component Analysis

Abstract

Robust tensor principal component analysis (RTPCA) based on tensor singular value decomposition (t-SVD) separates the low-rank component and the sparse component from the multiway data. For streaming data, online RTPCA (ORTPCA) processes tensor data sequentially, where the low-rank component is updated based on the latest estimation and the newly arrived sample. It enhances both computation and storage efficiency. However, in most of the existing ORTPCA methods, the relaxation from tensor multirank to the convex tensor nuclear norm (TNN) may have a certain modeling error, which leads to unavoidable tracking accuracy loss. In this article, a tensor Schatten-p norm ( $0\lt p\lt 1$ ) is applied to provide a tighter approximation of the tensor rank. A Lemma is deduced to divide the Schatten-p norm into terms to be updated in an online way. Based on it, the corresponding online nonconvex RTPCA (ONRTPCA) method is proposed for efficient tensor subspace tracking. Moreover, we incorporate the dynamic forgetting window into ONRTPCA to adaptively track varying subspaces. In addition, this article also provides convergence analysis and complexity analysis. Experimental results on synthetic data and real-world video data show that our proposed method achieves superior subspace tracking accuracy in comparison with a series of state-of-the-art methods while maintaining a high convergence speed and low memory requirement.