Heat-Kernel Entropy Profiles and Geometric Effective Sample Size for Weighted Measures on Manifolds

2026-07-07

AI Summary

This work addresses the limitation of traditional effective sample size (ESS) estimators, which disregard the geometric structure of the underlying support manifold and thus fail to capture the distributional characteristics of weighted measures in complex spaces. The authors propose a geometrically aware ESS metric—termed the heat kernel entropy profile—that uniquely integrates the heat kernel with Rényi entropy. By diffusing weighted atoms via intrinsic heat flow, the method tracks multiscale non-uniformities, preserving classical ESS properties while detecting proximate or redundant particles and intricate spherical structures. The approach combines heat kernel overlap computation, spherical harmonic decomposition, and self-normalized importance sampling, and establishes asymptotic theory on compact boundaryless manifolds. Theoretical analysis confirms its monotonicity, consistency, and asymptotic behavior, while spherical experiments successfully reveal geometric features—such as antipodal points, zonal bands, and multimodal configurations—that conventional methods overlook.

Abstract

Weighted empirical measures on compact manifolds arise in importance sampling, particle approximations, posterior summaries, quadrature, and representation learning. Standard weight-only summaries, such as ordinary effective sample size, ignore the geometry of the support. We introduce heat-kernel entropy profiles, a multiscale summary that diffuses weighted atoms by intrinsic heat flow and tracks nonuniformity across scales. For order-two Rényi entropy, the profile is computable from pairwise heat-kernel overlaps and yields a geometric effective sample size that discounts nearby or duplicate particles while matching ordinary effective sample size for well-separated particles. We prove monotonicity, small- and large-scale asymptotics, deterministic-weight consistency, and a bounded-ratio self-normalized importance-sampling extension for compact manifolds without boundary. On spheres, the unlogged profile decomposes into spherical-harmonic energies that recover mean-direction, von Mises-Fisher-type, and Bingham-type summaries. Sphere-based experiments show that the profile reveals antipodal, girdle, multimodal, and duplicate-particle structure missed by weight-only and first-moment spherical summaries.

Problem

• weighted measures
• effective sample size
• manifolds
• geometry
• heat kernel

Innovation

• heat-kernel entropy profiles
• geometric effective sample size
• weighted measures on manifolds
• Rényi entropy
• spherical harmonic decomposition