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

2026-07-09 · Source: stat.ML updates on arXiv.org · Field: Science & Research — Mathematics & Computational Sciences, Artificial Intelligence & Machine Learning · Depth: Expert, extended

Summary

The paper "Heat-Kernel Entropy Profiles and Geometric Effective Sample Size for Weighted Measures on Manifolds" introduces novel multiscale summaries for weighted empirical measures on compact Riemannian manifolds. These measures are common in importance sampling, particle approximations, and representation learning. The authors propose heat-kernel entropy profiles, which diffuse weighted atoms via intrinsic heat flow to track nonuniformity across scales. For order-two Rényi entropy, this profile is computable from pairwise heat-kernel overlaps and yields a geometric effective sample size (gESS). The gESS discounts nearby or duplicate particles, unlike ordinary effective sample size (ESS), while matching ESS for well-separated particles. The work proves monotonicity, small- and large-scale asymptotics, deterministic-weight consistency, and a bounded-ratio self-normalized importance-sampling extension. Experiments on spheres ($S^2$) demonstrate the profile's ability to reveal antipodal, girdle, multimodal, and duplicate-particle structures missed by weight-only and first-moment spherical summaries.

Key takeaway

For Research Scientists evaluating uncertainty in particle-based methods or representation learning on manifolds, you should consider adopting heat-kernel entropy profiles and geometric ESS. This approach provides a nuanced, scale-dependent view of particle distribution, revealing geometric structures like clusters or antipodal modes that traditional weight-only metrics like ordinary ESS miss. Incorporating gESS into your diagnostics will offer a more complete understanding of your weighted measures' effective sample size and occupied volume.

Key insights

Heat-kernel entropy profiles and geometric ESS provide multiscale, geometry-aware uncertainty summaries for weighted measures on manifolds.

Principles

• Geometric uncertainty is scale-dependent.
• Heat-kernel overlaps quantify particle similarity.
• Diffusion monotonically erases nonuniformity.

Method

Compute heat-kernel entropy profiles and gESS from pairwise heat-kernel overlaps (Gram matrices) by diffusing weighted atoms via intrinsic heat flow, then normalizing by local self-overlap.

In practice

• Diagnose weight degeneracy with geometric context.
• Identify multimodal or antipodal structures on spheres.
• Evaluate normalized embedding quality across scales.

Topics

• Heat-Kernel Entropy Profiles
• Geometric Effective Sample Size
• Riemannian Manifolds
• Importance Sampling
• Particle Methods
• Spherical Harmonics

Code references

• kisungyou/HeatKernelEntropyProfiles

Best for: AI Scientist, Research Scientist, Machine Learning Engineer

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Editorial summary, takeaway, and curation by AIssential. Original article published by stat.ML updates on arXiv.org.