Riemannian optimization on the simplex of positive definite matrices

Abstract

In this work, we generalize the probability simplex constraint to matrices, i.e., X_1 + X_2 + + X_K = I, where X_i 0 is a symmetric positive semidefinite matrix of size n n for all i = \1,,K \. By assuming positive definiteness of the matrices, we show that the constraint set arising from the matrix simplex has the structure of a smooth Riemannian submanifold. We discuss a novel Riemannian geometry for the matrix simplex manifold and show the derivation of first- and second-order optimization related ingredients.

Tasks

Reproductions