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.