基于热核熵的几何有效样本量，能捕捉加权粒子在流形上的分布不均。
Heat-Kernel Entropy Profiles and Geometric Effective Sample Size for Weighted Measures on Manifolds
- 用热核扩散在多尺度下衡量加权测度的非均匀性
- 首次提出几何有效样本量，可识别近邻或重复粒子
- 适用于重要性采样、后验近似等场景，对高维球面数据敏感
在紧致流形上，加权经验测度广泛应用于重要性采样、粒子近似、后验总结、求积及表征学习。传统有效样本量忽略支撑集的几何结构。本文引入热核熵轮廓，通过内在扩散在不同尺度上刻画非均匀性。对于二阶Rényi熵，成对热核重叠可给出精确轮廓与几何有效样本量（gESS），该量会降低近邻或重复粒子的贡献。当不同粒子间重叠趋于零时，gESS趋近于传统有效样本量。在无边紧致流形上，我们证明了轮廓单调性、gESS尺度极限、确定权重一致性以及自归一化重要性采样中的有界比结果。在球面上，未取对数的轮廓可分解为球谐能量，前几项对应vMF和Bingham型标量摘要。实验揭示了对跖、腰带、多模态及重复粒子结构，这些是仅依赖权重或一阶矩的摘要所遗漏的。
原文摘要 · Abstract (English)
Weighted empirical measures on compact manifolds appear in importance sampling, particle approximations, posterior summaries, quadrature, and representation learning. Ordinary effective sample size and related weight summaries ignore the geometry of the support. We introduce heat-kernel entropy profiles to measure nonuniformity after intrinsic diffusion at a range of scales. For order-two Rényi entropy, pairwise heat-kernel overlaps give an exact profile and a geometric effective sample size. This effective sample size discounts nearby or duplicate particles. It approaches ordinary effective sample size as overlaps between distinct particles vanish. On compact boundaryless manifolds, we establish profile monotonicity, gESS scale limits, deterministic-weight consistency, and a bounded-ratio result for self-normalized importance sampling. On spheres, the unlogged profile decomposes into spherical-harmonic energies. The first terms are squared mean-resultant and traceless-second-moment energies, which give vMF- and Bingham-type scalar summaries. Experiments identify antipodal, girdle, multimodal, and duplicate-particle structures that weight-only and first-moment summaries miss.
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