[Paper Review] Approximation of Riemannian measures by Stein's method

James Thompson|arXiv (Cornell University)|Jan 27, 2020

Geometric Analysis and Curvature Flows16 references4 citations

TL;DR

This paper develops a Riemannian extension of Stein's method for approximating probability measures on manifolds using semigroup representations and Bismut-type integration by parts formulas. It establishes Wasserstein distance bounds via derivative estimates of the solution to the Stein equation under curvature conditions, generalizing multivariate normal approximation to Riemannian settings with explicit dependence on Ricci curvature and Riemann curvature tensor bounds.

ABSTRACT

In this article, we present the theoretical basis for an approach to Stein's method for probability distributions on Riemannian manifolds. Using a semigroup representation for the solution to the Stein equation, we use tools from stochastic calculus to estimate the derivatives of the solution, yielding a bound on the Wasserstein distance. We first assume the Bakry-Emery-Ricci tensor is bounded below by a positive constant, after which we deal separately with the case of uniform approximation on a compact manifold. Applications of these results are currently under development and will appear in a subsequent article.

Motivation & Objective

• To extend Stein's method to Riemannian manifolds for distributional approximation.
• To address the lack of derivative estimation tools for the solution of the Stein equation on curved spaces.
• To establish quantitative Wasserstein distance bounds between a given distribution and a target measure on a Riemannian manifold.
• To generalize Fang, Shao, and Xu’s multivariate normal approximation to general Riemannian settings with curvature constraints.
• To provide a theoretical foundation for future applications in geometric probability and stochastic analysis on manifolds.

Proposed method

• Utilizes a semigroup representation of the solution to the Stein equation on a complete Riemannian manifold.
• Applies Bismut-type integration by parts formulas derived from local martingale techniques to estimate gradients, Hessians, and third derivatives of the semigroup.
• Employs stochastic calculus and Itô isometry to bound the derivatives of the solution using curvature assumptions.
• Derives explicit derivative estimates for the semigroup using Gronwall’s inequality under lower Ricci curvature bounds.
• Introduces a novel set of formulas for the first, second, and third derivatives of the semigroup via Cameron-Martin processes.
• Establishes bounds on the Wasserstein distance by combining curvature conditions with moment estimates on displacement and remainder terms.

Experimental results

Research questions

• RQ1 How can Stein’s method be generalized to Riemannian manifolds to approximate probability measures?
• RQ2 What curvature conditions are sufficient to control the derivatives of the solution to the Stein equation on a manifold?
• RQ3 Can Bismut-type formulas be extended to higher-order derivatives (Hessian and third derivative) of the semigroup?
• RQ4 What is the rate of convergence in Wasserstein distance for distributions on manifolds under curvature constraints?
• RQ5 How do the remainder terms in the approximation depend on the geometry of the manifold and the coupling of random variables?

Key findings

• Under a lower bound $ K > 0 $ on the Bakry-Emery-Ricci tensor $ \mathop{\rm Ric}_\psi \geq K $, the Wasserstein distance between a random variable $ W $ and the target measure $ \mu_\psi $ is bounded by $ \mathcal{W}(\mathcal{L}(W), \mu_\psi) \leq C \left( \frac{1}{\lambda} \mathbb{E}[|\delta|^3 (|\log|\delta|| \vee 1)] + \mathbb{E}[|R_1|] + \mathbb{E}[|R_2|] \right) $ for all $ \lambda > 0 $, where $ \delta = d(W, W') $.
• The bound includes a logarithmic term $ \log|\delta| $, which arises from the geometry of the manifold and is necessary for the Wasserstein control in non-flat settings.
• When the logarithmic term is removed, the bound is replaced by a $ C^2 $-distance, which is a weaker metric but avoids the $ \log|\delta| $ dependence.
• Derivative estimates for the semigroup are established: $ \|\nabla P_t f\|_\infty \leq e^{-Kt} \|\nabla f\|_\infty $, $ \|\nabla dP_t f\|_\infty \leq \frac{C_1 e^{-Kt}}{\sqrt{1 \wedge t}} \|\nabla f\|_\infty $, and $ \|\nabla\nabla dP_t f\|_\infty \leq \frac{C_2 e^{-Kt}}{1 \wedge t} \|\nabla f\|_\infty $, under curvature and boundedness assumptions.
• The results recover Fang, Shao, and Xu’s bound in the Euclidean case $ M = \mathbb{R}^n $, confirming consistency with existing multivariate normal approximation theory.
• The method relies on a complete set of Bismut-type formulas for first, second, and third derivatives of the semigroup, derived via local martingale techniques and Cameron-Martin processes.