  Tensor completion and robust principal component analysis have been widely
used in machine learning while the key problem relies on the minimization of a
tensor rank that is very challenging. A common way to tackle this difficulty is
to approximate the tensor rank with the $\ell_1-$norm of singular values based
on its Tensor Singular Value Decomposition (T-SVD). Besides, the sparsity of a
tensor is also measured by its $\ell_1-$norm. However, the $\ell_1$ penalty is
essentially biased and thus the result will deviate. In order to sidestep the
bias, we propose a novel non-convex tensor rank surrogate function and a novel
non-convex sparsity measure. In this new setting by using the concavity instead
of the convexity, a majorization minimization algorithm is further designed for
tensor completion and robust principal component analysis. Furthermore, we
analyze its theoretical properties. Finally, the experiments on natural and
hyperspectral images demonstrate the efficacy and efficiency of our proposed
method.
