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

2026-07-07 · Source: Machine Learning · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Data Science & Analytics, Emerging Technologies & Innovation · Depth: Expert, quick

Summary

Heat-kernel entropy profiles offer a multiscale summary for weighted empirical measures on compact manifolds, addressing the limitations of standard weight-only summaries like ordinary effective sample size, which ignore geometric support. This new approach diffuses weighted atoms using intrinsic heat flow, tracking nonuniformity across scales. For order-two Rényi entropy, the profile is computable from pairwise heat-kernel overlaps, yielding a geometric effective sample size that discounts nearby or duplicate particles while matching ordinary effective sample size for well-separated particles. The research proves monotonicity, small- and large-scale asymptotics, deterministic-weight consistency, and a bounded-ratio self-normalized importance-sampling extension for compact manifolds without boundary. Experiments on spheres demonstrate its ability to reveal complex structures—like antipodal, girdle, multimodal, and duplicate-particle arrangements—missed by traditional methods.

Key takeaway

For research scientists working with weighted empirical measures in areas like importance sampling or representation learning on compact manifolds, you should recognize that standard effective sample size metrics are geometrically blind. Consider adopting heat-kernel entropy profiles to gain a more accurate, multiscale understanding of particle distribution and nonuniformity. This method provides a geometric effective sample size, offering superior insights into structures like duplicate or clustered particles, which are crucial for robust model evaluation and development.

Key insights

Heat-kernel entropy profiles provide a geometry-aware effective sample size for weighted measures, improving upon traditional weight-only summaries.

Principles

• Standard effective sample size ignores geometric support.
• Heat flow diffusion tracks nonuniformity across scales.
• Geometric ESS discounts nearby particles.

Method

Compute heat-kernel entropy profiles using pairwise heat-kernel overlaps for order-two Rényi entropy to derive a geometric effective sample size.

In practice

• Analyze weighted measures in importance sampling.
• Improve particle approximations on manifolds.
• Summarize posterior distributions geometrically.

Topics

• Heat-Kernel Entropy Profiles
• Geometric Effective Sample Size
• Weighted Measures
• Compact Manifolds
• Importance Sampling
• Rényi Entropy

Best for: AI Scientist, Research Scientist

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Editorial summary, takeaway, and curation by AIssential. Original article published by Machine Learning.