  In order to compute the best low-rank tensor approximation using the
Multilinear Tensor Decomposition (MTD) model, it is essential to estimate the
rank of the underlying multilinear tensor from the noisy observation tensor. In
this paper, we propose a Robust MTD (R-MTD) method, which jointly estimates the
multilinear rank and the loading matrices. Based on the low-rank property and
an over-estimation of the core tensor, this joint estimation problem is solved
by promoting (group) sparsity of the over-estimated core tensor. Group sparsity
is promoted using mixed-norms. Then we establish a link between the mixed-norms
and the nuclear norm, showing that mixed-norms are better candidates for a
convex envelope of the rank. After several iterations of the Alternating
Direction Method of Multipliers (ADMM), the Minimum Description Length (MDL)
criterion computed from the eigenvalues of the unfolding matrices of the
estimated core tensor is minimized in order to estimate the multilinear rank.
The latter is then used to estimate more accurately the loading matrices. We
further develop another R-MTD method, called R-OMTD, by imposing an
orthonormality constraint on each loading matrix in order to decrease the
computation complexity. A series of simulated noisy tensor and real-world data
are used to show the effectiveness of the proposed methods compared with
state-of-the-art methods.
