Varadhan Functions, Variances, and Means on Compact Riemannian Manifolds

2026-01-06

AI Summary

This work addresses the asymptotic theory of Fréchet means and variances on compact Riemannian manifolds, which is typically hindered by the geometric complexity of the cut locus. To circumvent any assumptions on the cut locus, the authors introduce smooth surrogates—termed Varadhan functions, means, and variances—derived from the small-time asymptotics of the Varadhan heat kernel. By integrating tools from differential geometry, probability limit theory, and empirical process methods, they establish a uniform law of large numbers for the empirical versions of these smooth statistics and prove a central limit theorem under a fixed time parameter. Moreover, through a small-time analysis of the gradient and Hessian, they uncover a strong connection between the central limit theorems for Varadhan and Fréchet means, thereby offering new instruments for non-Euclidean statistical inference.

Technology Category

• Machine Learning: Learning with Manifolds
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Social Networks and Social Media: Social media analysis through the lenses of networks

Abstract

Motivated by Varadhan's theorem, we introduce Varadhan functions, variances, and means on compact Riemannian manifolds as smooth approximations to their Fr\'echet counterparts. Given independent and identically distributed samples, we prove uniform laws of large numbers for their empirical versions. Furthermore, we prove central limit theorems for Varadhan functions and variances for each fixed $t\ge0$, and for Varadhan means for each fixed $t>0$. By studying small time asymptotics of gradients and Hessians of Varadhan functions, we build a strong connection to the central limit theorem for Fr\'echet means, without assumptions on the geometry of the cut locus.

Problem

• Varadhan functions
• Fréchet means
• Riemannian manifolds
• central limit theorem
• large numbers law

Innovation

• Varadhan functions
• Fréchet means
• central limit theorem
• Riemannian manifolds
• small-time asymptotics