  In this paper, a novel approach via embedded tensor manifold regularization
for 2D+3D facial expression recognition (FERETMR) is proposed. Firstly, 3D
tensors are constructed from 2D face images and 3D face shape models to keep
the structural information and correlations. To maintain the local structure
(geometric information) of 3D tensor samples in the low-dimensional tensors
space during the dimensionality reduction, the $\ell_0$-norm of the core
tensors and a tensor manifold regularization scheme embedded on core tensors
are adopted via a low-rank truncated Tucker decomposition on the generated
tensors. As a result, the obtained factor matrices will be used for facial
expression classification prediction. To make the resulting tensor optimization
more tractable, $\ell_1$-norm surrogate is employed to relax $\ell_0$-norm and
hence the resulting tensor optimization problem has a nonsmooth objective
function due to the $\ell_1$-norm and orthogonal constraints from the
orthogonal Tucker decomposition. To efficiently tackle this tensor optimization
problem, we establish the first-order optimality condition in terms of
stationary points, and then design a block coordinate descent (BCD) algorithm
with convergence analysis and the computational complexity. Numerical results
on BU-3DFE database and Bosphorus databases demonstrate the effectiveness of
our proposed approach.
