Seminars

Abstract: A symmetric tensor is a higher order generalization of a symmetric matrix. It can be decomposed as a linear combination of simple symmetric tensors i.e. indecomposables symmetric tensors. This decomposition is called "the symmetric tensor decomposition" and the rank is the minimal number of terms in such a decomposition. We have a correspondence between the set of symmetric tensors of order d of dimension n and the set of homogeneous polynomials of degree d in n variables, which allows us to view the symmetric tensor decomposition problem as the decomposition of homogeneous polynomials into linear combination of the dth power of linear forms. In our approach, we introduce a Riemannian-Newton method to approximate this decomposition for homogeneous polynomials over the complex field, where we assume that the number of linear forms that is necessary to reconstruct this approximated decomposition is bounded by the generic rank. First, we formulate this problem as a Riemannian least square optimization problem i.e. a least square optimization problem such that the constraint set M is a Riemannian manifold. The parameterization that we consider yields a real valued objective function with real and complex variables which is a non-analytic function, to overcome this obstacle, we convert the optimization problem to the real domain by regarding f as a function of its real variables and the real and the imaginary parts of its complex variables, we denote this domain by R. second, we define another domian C of the real variables of f and the complex variables of f with their complex conjugate, this domain is in fact in isomorphism with R. We choose to compute the gradient and the Hessian of f which are the main components of a Newton iteration by regarding f as a function of C to obtain a simple elegant expressions, we do that using the apolar identities, then we use the isomorphism between R and C to define the expressions of the gradient and the Hessian of f as a function of R. We define a basis of the tangent space at a point on the Riemannian manifold M which allows as to define the expression of the Riemannian Newton equation. Third, a new retraction operator from the tangent space at a point of the Veronese manifold to the Veronese manifold is proposed, using the SVD decomposition of the Hankel matrix obtained by flatting the tangent vector, numerical experiments show that this new retraction has approximately the same performance as the retraction by the projection on the sphere. In our approach we do all possible simplification to get a new iterative methode which is on the one hand a computationally effective compact method and on the other hand preserve the desirable quadratic rate of convergence. Finally,  numerical experiments show the efficiency and the numerical quality of the method. More numerical experiences can gives us insights for the best rank approximation in some cases by taking in consideration the scale of perturbation, the choice of the initial point of the algorithm and some others parameters.

Keywords: Symmetric tensor, symmetric tensor decomposition, homogeneous polynomials, Riemannian manifolds, Riemannian optimization, complex optimization, retraction.