Riemannian optimization on the simplex of positive definite matrices

In this work, we generalize the probability simplex constraint to matrices, i.e., X1 + X2 + ldots + XK = I, where Xi succeq 0 is a symmetric positive semidefinite matrix of size n× n for all i = 1,ldots,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.