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 q-way arrays (q>2), that is, the arrays whose elements are addressed by more than two indices. The first works dates back to Jordan who was interested in simultaneously diagonalizing two matrices at a time [93] . It is noteworthy that such two matrices can be interpreted as both slices of a three-way array and their joint diagonalization can be viewed as Hitchcock's polyadic decomposition [89] of the associated three-way array. Other works followed discussing rank problems related to multi-way structures and properties of multi-way arrays. However, these exercices in multilinear algebra were not linked to real data analysis but stayed within the realm of mathematics. Studying three-way data really started with Tucker's seminal work, which gave birth to the three-mode factor analysis [112] . His model is now often referred to as the Tucker3 model. At the same moment, other authors focused on a particular case of the Tucker3 model, calling it PARAFAC for PARAllel FACtor analysis [88] , and on the means to achieve such a decomposition, which will become the famous canonical decomposition [73] . In honor to Hitchcock's pionneer work, we will call it the Canonical Polyadic (CP) decomposition.

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 𝐋𝐔 factorization of the matrix of eigenvectors [98] . The JEVD can be useful in several contexts such as CP decomposition of tensors [99] and more particularly in Independent Component Analysis (ICA) based on higher order cumulants where it allows us to blindly compute the mixing matrix of sources with kurtosis of different signs. Regarding the proposed JET approach, contrary to classical Jacobi-like JEVD methods, its iterative procedure can be reduced to the search for only one of the two triangular matrices involved in the factorization of the matrix of eigenvectors, hence decreasing the numerical complexity. Two variants of the JET technique, namely JET-U and JET-O, which correspond to the optimization of two different cost functions were described in detail and these were extended to the complex case. Numerical simulations showed that in many practical cases the JET approach provides more accurate estimation of the matrix of eigenvectors than its competitors and that the lowest numerical complexity is consistently achieved by the JET-U algorithm.