  In this paper, we propose a new approach to solve low-rank tensor completion
and robust tensor PCA. Our approach is based on some novel notion of
(even-order) tensor ranks, to be called the M-rank, the symmetric M-rank, and
the strongly symmetric M-rank. We discuss the connections between these new
tensor ranks and the CP-rank and the symmetric CP-rank of an even-order tensor.
We show that the M-rank provides a reliable and easy-computable approximation
to the CP-rank. As a result, we propose to replace the CP-rank by the M-rank in
the low-CP-rank tensor completion and robust tensor PCA. Numerical results
suggest that our new approach based on the M-rank outperforms existing methods
that are based on low-n-rank, t-SVD and KBR approaches for solving low-rank
tensor completion and robust tensor PCA when the underlying tensor has low
CP-rank.
