Measure estimation on a manifold explored by a diffusion process

TL;DR

This paper investigates the estimation of a stationary measure on a manifold from diffusion paths, extending previous results to broader classes of diffusions and demonstrating faster convergence rates when the density's regularity is leveraged.

Contribution

The authors extend convergence rate results to a wider class of diffusion paths and establish faster estimators using density regularity, achieving minimax optimal rates.

Findings

Convergence rate of $T^{-1/(d-2)}$ for a broad class of diffusions.

Smoothing estimators improve convergence rates based on density regularity.

Proved that the new rates are minimax optimal.

Abstract

From the observation of a diffusion path $(X_t)_{t\in [0,T]}$ on a compact connected $d$-dimensional manifold $\mathcal{M}$ without boundary, we consider the problem of estimating the stationary measure $\mu$ of the process. Wang and Zhu (2023) showed that for the Wasserstein metric $\mathcal{W}_2$ and for $d\geq 5$, the convergence rate of $T^{-1/(d-2)}$ is attained by the occupation measure of the path $(X_t)_{t\in [0,T]}$ when $(X_t)_{t\in [0,T]}$ is a Langevin diffusion. We extend their result in several directions. First, we show that the rate of convergence holds for a large class of diffusion paths, whose generators are uniformly elliptic. Second, the regularity of the density $p$ of the stationary measure $\mu$ with respect to the volume measure of $\mathcal{M}$ can be leveraged to obtain faster estimators: when $p$ belongs to a Sobolev space of order $\ell\geq 2$, smoothing the occupation measure by convolution with a kernel yields an estimator whose rate of convergence is of order $T^{-(\ell+1)/(2\ell+d-2)}$. We further show that this rate is the minimax rate of estimation for this problem.