Riemannian Newton optimization methods for the symmetric tensor approximation problem

The Symmetric Tensor Approximation problem (STA) consists of approximating a symmetric tensor or homogeneous polynomial by linear combination rank-1 tensors powers forms low rank. We present two new Riemannian Newton-type methods for rank approximation with complex coefficients. first method uses the parametrization set at most r weights and unit vectors. Exploiting properties apolar product on polynomials combined efficient tools from optimization, we provide an explicit tractable formulation gradient Hessian, leading to Newton iterations local quadratic convergence. prove that under some regularity conditions non-defective in neighborhood initial point, iteration (completed trust-region scheme) is converging minimum. second Gauss–Newton Cartesian Veronese manifolds. An orthonormal basis tangent space this manifold described. deduce Hessian. retraction operator manifold. analyze numerical behavior these methods, point provided Simultaneous Matrix Diagonalisation (SMD). Numerical experiments show good different cases comparison existing state-of-the-art methods.