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Measure estimation on a manifold explored by a diffusion process

Vincent Divol, Hélène Guérin, Dinh-Toan Nguyen, Viet Chi Tran

TL;DR

This work addresses estimating the invariant measure of a diffusion on a compact manifold from a single diffusion path. It introduces a kernel-smoothed occupation measure estimator $\widehat{\mu}_{T,h}$ and proves nonasymptotic upper bounds in the Wasserstein-2 metric that balance a variance term (scaling like $h^{4-d}/T$ for $d\ge5$) and a bias term tied to the density's Sobolev regularity $\ell$. When the stationary density $p$ lies in Sobolev space $H^\ell(\mathcal{M})$ with $\ell\ge2$, choosing $h\asymp T^{-1/(2\ell+d-2)}$ yields an overall rate of $\mathcal{O}\big(T^{-(2\ell+2)/(2\ell+d-2)}\big)$, which is minimax optimal in this regime. A minimax lower bound shows these rates are optimal for $d\ge5$ (with a slower rate for $d\le4$ where the empirical occupation measure can be competitive). The analysis hinges on elliptic operator theory on manifolds (Laplace–Beltrami, Green function), ultracontractivity, and a careful variance-bias decomposition, with results that extend beyond Langevin diffusions to a broad class of uniformly elliptic generators $\mathcal{A}$. This provides a principled, nonparametric route to density estimation on manifolds from diffusion data, with explicit finite-sample guarantees.

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 $μ$ 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 $μ$ 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.

Measure estimation on a manifold explored by a diffusion process

TL;DR

This work addresses estimating the invariant measure of a diffusion on a compact manifold from a single diffusion path. It introduces a kernel-smoothed occupation measure estimator and proves nonasymptotic upper bounds in the Wasserstein-2 metric that balance a variance term (scaling like for ) and a bias term tied to the density's Sobolev regularity . When the stationary density lies in Sobolev space with , choosing yields an overall rate of , which is minimax optimal in this regime. A minimax lower bound shows these rates are optimal for (with a slower rate for where the empirical occupation measure can be competitive). The analysis hinges on elliptic operator theory on manifolds (Laplace–Beltrami, Green function), ultracontractivity, and a careful variance-bias decomposition, with results that extend beyond Langevin diffusions to a broad class of uniformly elliptic generators . This provides a principled, nonparametric route to density estimation on manifolds from diffusion data, with explicit finite-sample guarantees.

Abstract

From the observation of a diffusion path on a compact connected -dimensional manifold without boundary, we consider the problem of estimating the stationary measure of the process. Wang and Zhu (2023) showed that for the Wasserstein metric and for , the convergence rate of is attained by the occupation measure of the path when 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 of the stationary measure with respect to the volume measure of can be leveraged to obtain faster estimators: when belongs to a Sobolev space of order , smoothing the occupation measure by convolution with a kernel yields an estimator whose rate of convergence is of order . We further show that this rate is the minimax rate of estimation for this problem.

Paper Structure

This paper contains 23 sections, 31 theorems, 187 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Preliminaries
  4. The Laplace-Beltrami operator
  5. Green function of the Laplace Beltrami operator
  6. The elliptic operator $\mathcal{A}$
  7. Variance term for the stationary process
  8. Estimation of the probability ${\mathbf P}_\mu(E_{T,h}^c)$
  9. Proof of \ref{['prop:variance']}
  10. Transition to a general initial measure
  11. Minimax lower bound
  12. Wasserstein distance and bias term estimates
  13. About the Wasserstein distance's estimate of Peyre and Proof of Lemma \ref{['lem:wang']}
  14. Control of the bias term
  15. Some technical proofs
  16. ...and 8 more sections

Key Result

Theorem 2.3

Let $d\ge 1$ and $p$ be a positive $\mathcal{C}^2$ density function with associated measure $\mu$. Let $(X_t)_{t\ge 0}$ be a diffusion with generator $\mathcal{A}$ satisfying Assumption assump:A. Let $T\ge 2$ and let $0<h\le h_0$ for some constant $h_0$ depending on $\mathcal{M}$ and $K$. Assume tha where $c_0$ depends on $\mathcal{M}$, and $c$ depends on $\mathcal{M}$, $K$, $p_{\min}$, $p_{\max}$

Theorems & Definitions (63)

  • Remark 2.2
  • Theorem 2.3: Estimation from a diffusion with generator $\mathcal{A}$
  • Corollary 2.4
  • Corollary 2.5
  • proof
  • Proposition 2.6
  • Remark 2.7
  • Remark 2.8
  • Proposition 3.1: See Appendix A in Alesker2013, or Theorem 4.13 in Aubin1982
  • Lemma 3.2
  • ...and 53 more