Riemannian optimization on the simplex of positive definite matrices
This work addresses optimization on matrix simplex constraints, which is incremental as it extends existing simplex methods to matrices.
The paper tackles the problem of generalizing the probability simplex constraint to matrices, showing that the constraint set forms a smooth Riemannian submanifold and deriving first- and second-order optimization ingredients for it.
In this work, we generalize the probability simplex constraint to matrices, i.e., $\mathbf{X}_1 + \mathbf{X}_2 + \ldots + \mathbf{X}_K = \mathbf{I}$, where $\mathbf{X}_i \succeq 0$ is a symmetric positive semidefinite matrix of size $n\times n$ 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.