Section: New Results
Recent results on tensor decompositions
tensor, multiway array, canonical polyadic decomposition, nonnegative tensor factorization
Multi-linear algebra is defined as the algebra of
Achieving a CP decomposition has been seen first as a mere non-linear least squares problem, with a simple objective criterion. In fact, the objective is a polynomial function of many variables, where some separate. One could think that this kind of objective is easy because smooth, and even infinitely differentiable. But it turns out that things are much more complicated than they may appear to be at first glance. Nevertheless, the Alternating Least Squares (ALS) algorithm has been mostly utilized to address this minimization problem, because of its programming simplicity. This should not hide the inherently complicated theory that lies behind the optimization problem. Moreover, in most of the applications, actual tensors may not exactly satisfy the expected model, so that the problem is eventually an approximation rather than an exact decomposition. This may result in a slow convergence (or lack of convergence) of iterative algorithms such as the ALS one [97] . Consequently, a new class of efficient algorithms able to take into account the properties of tensors to be decomposed is needed.
CP decomposition of semi-symmetric three-way arrays subject to arbitrary convex constraints
Participant : Laurent Albera.
Main collaborations : Lu Wang (LTSI, France), Amar Kachenoura (LTSI, France), Lotfi Senhadji (LTSI, France), Jean-Christophe Pesquet (LIGM, France)
We addressed the problem of canonical polyadic decomposition of semi-symmetric 3rd order tensors (i.e. joint diagonalization by congruence) subject to arbitrary convex constraints. Sufficient conditions for the existence of a solution were proved. An efficient algorithm based on the Alternating Direction Method of Multipliers (ADMM) was then designed. ADMM provides an elegant approach for handling the additional constraint terms, while taking advantage of the structure of the objective function. Numerical tests on simulated matrices showed the benefits of the proposed method for low signal to noise ratios. Simulations in the context of nuclear magnetic resonance spectroscopy were also provided. This work was presented at the IEEE CAMSAP'15 conference [29] .
Joint eigenvalue decomposition of non-defective matrices for the CP decomposition of tensors
Participant : Laurent Albera.
We proposed a fast and efficient Jacobi-like approach named JET (Joint Eigenvalue decomposition based on Triangular matrices) for the Joint EigenValue Decomposition (JEVD) of a set of real or complex non-defective matrices based on the