Riemannian optimization on the simplex of positive definite matrices

Abstract

In this work, we generalize the probability simplex constraint to matrices, i.e., ${\mathbf{X}}_1+{\mathbf{X}}_2+\dots +{\mathbf{X}}_K=\mathbf{I}$, where ${\mathbf{X}}_i⪰0$ is a symmetric positive semidefinite matrix of size $n\times n$ for all $i=\{1,\dots ,K\}$. 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.

12th OPT Workshop on Optimization for Machine Learning at NeurIPS 2020