Robust Low-tubal-rank Tensor Recovery from Binary Measurements
Jingyao Hou, Feng Zhang, Haiquan Qiu, Jianjun Wang, Yao Wang, Deyu Meng
Abstract
Low-rank tensor recovery (LRTR) is a natural extension of low-rank matrix recovery (LRMR) to high-dimensional arrays, which aims to reconstruct an underlying tensor from incomplete linear measurements M(X). However, LRTR ignores the error caused by quantization, limiting its application when the quantization is low-level. In this work, we take into account the impact of extreme quantization and suppose the quantizer degrades into a comparator that only acquires the signs of M(X). We still hope to recover X from these binary measurements. Under the tensor Singular Value Decomposition (t-SVD) framework, two recovery methods are proposedthe first is a tensor hard singular tube thresholding method; the second is a constrained tensor nuclear norm minimization method. These methods can recover a real n1 n2 n3 tensor X with tubal rank r from m random Gaussian binary measurements with errors decaying at a polynomial speed of the oversampling factor := m/((n1+ n2)n3r). To improve the convergence rate, we develop a new quantization scheme under which the convergence rate can be accelerated to an exponential function of . Numerical experiments verify our results, and the applications to real-world data demonstrate the promising performance of the proposed methods.