Line search and trust region strategies for canonical decomposition of semi-nonnegative semi-symmetric 3rd order tensors

Julie Coloigner (1) , Ahmad Karfoul (2) , Laurent Albera (1, 3) , Pierre Comon (4)

1 LTSI - Laboratoire Traitement du Signal et de l'Image
2 Faculty of Mechanical and Electrical Engineering [Homs]
3 PANAMA - Parcimonie et Nouveaux Algorithmes pour le Signal et la Modélisation Audio
4 GIPSA-CICS - GIPSA - Communication Information and Complex Systems

Numerical solutions are proposed to fit the CanDecomp/ParaFac (CP) model of real three-way arrays, when the latter are both nonnegative and symmetric in two modes. In other words, a semi- nonnegative INDSCAL analysis is performed. The nonnegativity constraint is circumvented by means of changes of variable into squares, leading to an unconstrained problem. In addition, two globalization strategies are studied, namely line search and trust region. Regarding the former, a global plane search scheme is considered. It consists in computing, for a given direction, one or two optimal stepsizes, depending on whether the same stepsize is used in various updating rules. Moreover, we provide a compact matrix form for the derivatives of the objective func- tion. This allows for a direct implementation of several iterative algorithms such as conjugate gradient, Levenberg-Marquardt and Newton-like methods, in matrix programming environments like MATLAB. Our numerical results show the advantage of our optimization strategies when combined with a priori information such as partial symmetry.

Mots clés: matrix derivation; Canonical polyadic model; nonnegative matrix factorization; nonnegative tensor factorization; semi-nonnegative INDSCAL analysis; CanDecomp; ParaFac; nonlinear optimization; matrix derivation.

Domaines: Analyse numérique [math.NA]; Traitement du signal et de l'image [eess.SP]; Traitement du signal et de l'image [eess.SP]

Julie Coloigner, Ahmad Karfoul, Laurent Albera, Pierre Comon. Line search and trust region strategies for canonical decomposition of semi-nonnegative semi-symmetric 3rd order tensors. Linear Algebra and its Applications, 2014, 450, pp.334-374. ⟨10.1016/j.laa.2014.02.001⟩. ⟨hal-00945606⟩