Kisung You's Heat-Kernel Entropy Gives Geometric Effective Sample Size on Manifolds

Kisung You's new paper on arXiv (2607.06696) tackles a fundamental problem in statistics and machine learning: how to summarize weighted empirical measures on curved spaces like spheres, tori, or more general compact manifolds. Standard weight-only summaries, such as the ordinary effective sample size (ESS), ignore the geometry of the support—meaning particles sitting right next to each other get the same weight credit as particles far apart. You's solution uses heat-kernel entropy profiles: at multiple scales, the method diffuses weighted atoms using the manifold's intrinsic heat flow and tracks nonuniformity via order-two Rényi entropy. The resulting geometric effective sample size automatically discounts nearby or duplicate particles while reverting to ordinary ESS for well-separated ones. The paper proves monotonicity, small- and large-scale asymptotics, and consistency for self-normalized importance sampling on compact manifolds without boundary.

On spheres specifically, the unlogged profile decomposes into spherical-harmonic energies that recover mean-direction, von Mises-Fisher-type, and Bingham-type summaries—classical tools for directional statistics. In experiments, the profile reveals antipodal, girdle, multimodal, and duplicate-particle structures that weight-only and first-moment spherical summaries completely miss. This work has immediate applications in importance sampling, particle filtering, quadrature on spheres, and representation learning, where geometric awareness can dramatically improve sample efficiency. For practitioners using directional statistics or manifold-constrained Bayesian inference, this new metric offers a principled way to diagnose particle degeneracy and design better sampling strategies.