Approximating Symmetric Positive Semidefinite Tensors of Even Order

Tensors of various orders can be used for modeling physical quantities such as strain and diffusion as well as curvature and other quantities of geometric origin. Depending on the physical properties of the modeled quantity, the estimated tensors are often required to satisfy the positivity constraint, which can be satisfied only with tensors of even order. Although the space P-0(2m) of 2mth-order symmetric positive semidefinite tensors is known to be a convex cone, enforcing positivity constraint directly on P-0(2m) is usually not straightforward computationally because there is no known analytic description of P-0(2m) for m > 1. In this paper, we propose a novel approach for enforcing the positivity constraint on even-order tensors by approximating the cone P-0(2m) for the cases 0 < m < 3, and presenting an explicit characterization of the approximation Sigma(2m) subset of Omega(2m) for m >= 1, using the subset Omega(2m) subset of P-0(2m) of semidefinite tensors that can be written as a sum of squares of tensors of order m. Furthermore, we show that this approximation leads to a nonnegative linear least-squares optimization problem with the complexity that equals the number of generators in Sigma(2m). Finally, we experimentally validate the proposed approach and present an application for computing 2mth-order diffusion tensors from diffusion weighted magnetic resonance images.