  The goal of this paper is to find a low-rank approximation for a given
tensor. Specifically, we give a computable strategy on calculating the rank of
a given tensor, based on approximating the solution to an NP-hard problem. In
this paper, we formulate a sparse optimization problem via an
$l_1$-regularization to find a low-rank approximation of tensors. To solve this
sparse optimization problem, we propose a rescaling algorithm of the proximal
alternating minimization and study the theoretical convergence of this
algorithm. Furthermore, we discuss the probabilistic consistency of the
sparsity result and suggest a way to choose the regularization parameter for
practical computation. In the simulation experiments, the performance of our
algorithm supports that our method provides an efficient estimate on the number
of rank-one tensor components in a given tensor. Moreover, this algorithm is
also applied to surveillance videos for low-rank approximation.
