  Higher-order low-rank tensor arises in many data processing applications and
has attracted great interests. Inspired by low-rank approximation theory,
researchers have proposed a series of effective tensor completion methods.
However, most of these methods directly consider the global low-rankness of
underlying tensors, which is not sufficient for a low sampling rate; in
addition, the single nuclear norm or its relaxation is usually adopted to
approximate the rank function, which would lead to suboptimal solution deviated
from the original one. To alleviate the above problems, in this paper, we
propose a novel low-rank approximation of tensor multi-modes (LRATM), in which
a double nonconvex $L_{\gamma}$ norm is designed to represent the underlying
joint-manifold drawn from the modal factorization factors of the underlying
tensor. A block successive upper-bound minimization method-based algorithm is
designed to efficiently solve the proposed model, and it can be demonstrated
that our numerical scheme converges to the coordinatewise minimizers. Numerical
results on three types of public multi-dimensional datasets have tested and
shown that our algorithm can recover a variety of low-rank tensors with
significantly fewer samples than the compared methods.

Computer Vision and Pattern Recognition