Learnable Scaled Gradient Descent for Guaranteed Robust Tensor PCA

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.

Frequency-Weighted Robust Tensor Principal Component Analysis

Robust tensor principal component analysis (RTPCA) can separate the low-rank component and sparse component from multidimensional data, which has been used successfully in several image applications. Its performance varies with different kinds of tensor decompositions, and the tensor singular value decomposition (t-SVD) is a popularly selected one. The standard t-SVD takes the discrete Fourier transform to exploit the residual in the 3rd mode in the decomposition. When minimizing the tensor nuclear norm related to t-SVD, all the frontal slices in frequency domain are optimized equally. In this paper, we incorporate frequency component analysis into t-SVD to enhance the RTPCA performance. Specially, different frequency bands are unequally weighted with respect to the corresponding physical meanings, and the frequency-weighted tensor nuclear norm can be obtained. Accordingly we rigorously deduce the frequency-weighted tensor singular value threshold operator, and apply it for low rank approximation subproblem in RTPCA. The newly obtained frequency-weighted RTPCA can be solved by alternating direction method of multipliers, and it is the first time that frequency analysis is taken in tensor principal component analysis. Numerical experiments on synthetic 3D data, color image denoising and background modeling verify that the proposed method outperforms the state-of-the-art algorithms both in accuracy and computational complexity.