# Riemannian optimization on the simplex of positive definite matrices

arXiv: https://arxiv.org/abs/1906.10436 · HTML: https://mlsanity.com/paper/1906.10436/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 Jun 2019.

## Built on by (most cited first, 4 of 4)

- [On Riemannian Optimization over Positive Definite Matrices with the Bures-Wasserstein Geometry](https://mlsanity.com/paper/2106.00286/) (2021)
- [Riemannian block SPD coupling manifold and its application to optimal transport](https://mlsanity.com/paper/2201.12933/) (2022)
- [Mixed-curvature decision trees and random forests](https://mlsanity.com/paper/2410.13879/) (2024)
- [Mixed-Curvature Decision Trees and Random Forests](https://mlsanity.com/paper/2406.05227/) (2024)

## Abstract (arXiv)

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.