We are interested here in one of the most popular tensor factorization namely the Canonical Polyadic Decomposition (CPD) for N-th order tensors. It consists in decomposing a tensor into a minimal sum of rank-one terms, the number of rank-one tensors being the tensor rank (possibly unknown). The tensor satisfies a multi-linear property with respect to its loading factors. The objective of this work is to recover the loading factors that define the observed tensor modeled by the CP model. The large size of the observations as well as possible perturbations and the lack of knowledge of the actual tensor rank make the CPD inverse problem often difficult to solve. We thus propose to formulate the CP problem as an optimization problem to be solved, the criterion to be minimized gathering a data fidelity term involving the observed tensor and the unknow loading factors and penalizations acting individually on the loading factors (possibly favoring sparsity and ensuring nonnegativity). Then, based on a block coordinate variable metric forward-backward (BC-VMFB) method, we proposed a new penalized nonnegative N-th order CPD algorithm. It consists in four main steps : i) a gradient step in the forward stage, ii) a proximal step in the backward stage, iii) a preconditioning step (“variable metric”), and iv) a block arrangement (“Block Coordinate”) of the unknown (latent) variables that will be generally swept according to a random or cyclic rule . The efficiency (flexibility, robustness and speed) of the proposed algorithm is not only demonstrated through synthetic third and fourth order tensor decomposition but also on real 3D fluorescence spectroscopy data (water monitoring experiment).

This presentation is part of Minisymposium “MS10 - Advanced optimization methods for image processing (2 parts)”
organized by: Marco Prato (University of Modena and Reggio Emilia) , Ignace Loris (Université Libre de Bruxelles) .

Authors:

Caroline Chaux (Aix-Marseille Université)

Xuan Vu (Aix-Marseille Université)

Nadège Thirion-Moreau (Aix-Marseille Université)

Sylvain Maire (Aix-Marseille Université)

Keywords:

inverse problems, optimization, tensor factorization