arXiv:2003.01469v1 [math.NA] 3 Mar 2020 A RIEMANNIAN NEWTON OPTIMIZATION FRAMEWORK FOR THE SYMMETRIC TENSOR RANK APPROXIMATION PROBLEM RIMA KHOUJA †‡ , HOUSSAM KHALIL * , AND BERNARD MOURRAIN † Abstract. The symmetric tensor rank approximation problem (STA) consists in computing the best low rank approximation of a symmetric tensor. We describe a Riemannian Newton iteration with trust region scheme for the STA problem. We formulate this problem as a Riemannian optimization problem by parameterizing the constraint set as the Cartesian product of Veronese manifolds. We present an explicit and exact formula for the gradient vector and the Hessian matrix of the method, in terms of the weights and points of the low rank approximation and the symmetric tensor to approximate, by exploiting the properties of the apolar product. We introduce a retraction operator on the Veronese manifold. The Newton Riemannian iterations are performed for best low rank approximation over the real or complex numbers. Numerical experiments are implemented to show the numerical behavior of the new method first against perturbation, to compute the best rank-1 approximation and the spectral norm of a symmetric tensor, and to compare with some existing state-of-the-art methods. Key words. symmetric tensor decomposition, homogeneous polynomials, Riemannian optimiza- tion, Newton method, retraction, complex optimization, trust region method, Veronese manifold. AMS subject classifications. 15A69, 15A18, 53B20, 53B21, 14P10, 65K10, 65Y20, 90-08 1. Introduction. A symmetric tensor T of order d and dimension n in T d (C n )= C n ⊗···⊗ C n := T d n , is a special case of tensors, where its entries do not change under any permutation of its d indices. We denote their set S d (C n ) := S d n . The Symmetric Tensor Decomposition problem (STD) consists in decomposing the symmetric tensor into linear combination of symmetric tensors of rank one i.e. (STD) T = r  i=1 w i v i ⊗ ... ⊗ v i    d times , w i ∈ C,v i ∈ C n The smallest r such that this decomposition exists is by definition the symmetric rank of P . The STD appears in many applications in the areas of the mobile com- munications, in blind identification of under-determined mixtures, machine learning, factor analysis of k-way arrays, statistics, biomedical engineering, psychometrics, and chemometrics. See e.g. [14, 16, 17, 34] and references therein. The decomposition of the tensor is often used to recover structural information in the application problem. The Symmetric Tensor Approximation problem (STA) consists in finding the closest symmetric tensor ∈S d n , which has a symmetric rank at most r for a given r ∈ N. Using the correspondence between S d n and the set of homogeneous polynomials of degree d in n variables denoted C[x 1 ,...,x n ] d := C[x] d , it consists in approximating an homogeneous polynomial P associated to a symmetric tensor T by an element in σ r , where σ r = {Q ∈ C[x] d | rank s (Q) ≤ r}, i.e. (STA) min Q∈σr 1 2 ||P − Q|| 2 d . Since in many problems, the input tensors are often computed from measurements or statistics, they are known with some errors on their coefficients and computing an * Laboratory of Mathematics and its Applications LaMa-Lebanon, Lebanese University, Faculty of Sciences, Lebanon. (houssam.khalil@ul.edu.lb). † Aromath, Inria Sophia Antipolis M´ editerran´ ee, Universit´ e Cˆ ote d’Azur, France. (rima.khouja@inria.fr, bernard.mourrain@inria.fr). 1