  Dictionary Learning (DL) is one of the leading sparsity promoting techniques
in the context of image classification, where the "dictionary" matrix D of
images and the sparse matrix X are determined so as to represent a redundant
image dataset. The resulting constrained optimization problem is nonconvex and
non-smooth, providing several computational challenges for its solution. To
preserve multidimensional data features, various tensor DL formulations have
been introduced, adding to the problem complexity. We develop a new alternating
algorithm for the solution of the DL problem both in the matrix and tensor
frameworks; in the latter case a new formulation based on Tensor-Train
decompositions is also proposed. The new method belongs to the Proximal
Alternating Linearized Minimization (PALM) algorithmic family, with the
inclusion of second order information to enhance efficiency. We discuss a
rigorous convergence analysis, and report on the new method performance on the
image classification of several benchmark datasets.
