# The Riemannian Langevin equation and conic programs

## TL;DR

This paper introduces the Riemannian Langevin equation (RLE) as a generalization of stochastic gradient methods to Riemannian manifolds, providing explicit formulas for Brownian motion on cones, advancing understanding of stochastic processes in geometric spaces.

## Contribution

It formulates the Riemannian Langevin equation and derives explicit formulas for Brownian motion on fundamental cones, expanding stochastic analysis on manifolds.

## Key findings

- Explicit formulas for Brownian motion on cones
- Generalization of Langevin dynamics to Riemannian manifolds
- Framework for analyzing stochastic processes on geometric spaces

## Abstract

Diffusion limits provide a framework for the asymptotic analysis of stochastic gradient descent (SGD) schemes used in machine learning. We consider an alternative framework, the Riemannian Langevin equation (RLE), that generalizes the classical paradigm of equilibration in R^n to a Riemannian manifold (M^n, g). The most subtle part of this equation is the description of Brownian motion on (M^n, g). Explicit formulas are presented for some fundamental cones.