A Tensor-Train Dictionary Learning algorithm based on Spectral Proximal Alternating Linearized Minimization

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 propose a new tensor formulation of the DL problem using a Tensor-Train decomposition of the multi-dimensional dictionary, together with a new alternating algorithm for its solution. 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.