Robust Low-Tubal-Rank Tensor Recovery From Binary Measurements

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 $\boldsymbol{\mathcal {X}}$ from incomplete linear measurements $\mathfrak {M}(\boldsymbol{\mathcal {X}})$ . 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 $\mathfrak {M}(\boldsymbol{\mathcal {X}})$ . We still hope to recover $\boldsymbol{\mathcal {X}}$ from these binary measurements. Under the tensor Singular Value Decomposition (t-SVD) framework, two recovery methods are proposed—the 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 $n_1\times n_2\times n_3$ tensor $\boldsymbol{\mathcal {X}}$ with tubal rank $r$ from $m$ random Gaussian binary measurements with errors decaying at a polynomial speed of the oversampling factor $\lambda :=m/((n_1+n_2)n_3r)$ . To improve the convergence rate, we develop a new quantization scheme under which the convergence rate can be accelerated to an exponential function of $\lambda$ . Numerical experiments verify our results, and the applications to real-world data demonstrate the promising performance of the proposed methods.