Low tubal rank tensor sensing and robust PCA from quantized measurements

讲座时间 Datetime: 2022年9月13日，星期二，19:00-21:00

地点 Venue: 腾讯会议：758-640-599

主持人 Host：刘海峰

报告人 Speaker: 王建军  

单位 Affiliation: 西南大学

报告摘要 Abstract:

Low-rank tensor Sensing (LRTS) is a natural extension of low-rank matrix Sensing (LRMS) to high-dimensional arrays, which aims to reconstruct an underlying tensor X from incomplete linear measurements M(X). However, LRTS ignores the error caused by quantization, limiting its application when the quantization is low-level. Under the tensor Singular Value Decomposition (t-SVD) framework, two recovery methods are proposed. These methods can recover a real tensor X with tubal rank r from m random Gaussian binary measurements with errors decaying at a polynomial speed of the oversampling factor. To improve the convergence rate, we develop a new quantization scheme under which the convergence rate can be accelerated to an exponential function of lambda. Quantized Tensor Robust Principal Component Analysis (Q-TRPCA) aims to recover a low-rank tensor and a sparse tensor from noisy, quantized, and sparsely corrupted measurements. A nonconvex constrained maximum likelihood (ML) estimation method is proposed for Q-TRPCA. We provide an upper bound on the Frobenius norm of tensor estimation error under this method. Making use of tools in information theory, we derive a theoretical lower bound on the best achievable estimation error from unquantized measurements. Compared with the lower bound, the upper bound on the estimation error is nearly order-optimal. We further develop an efficient convex ML estimation scheme for Q-TRPCA based on the tensor nuclear norm (TNN) constraint. This method is more robust to sparse noises than the latter nonconvex ML estimation approach. Numerical experiments verify our results, and the applications to real-world data demonstrate the promising performance of the proposed methods.