发表机构
Baruch College（巴克尔学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究紧致流形上加权测度，引入热核熵剖面这一多尺度总结方法，通过内在热流扩散加权原子并跟踪尺度不均匀性，可计算二阶Rényi熵的几何有效样本量，实验表明其能揭示传统总结遗漏的结构。
AI 中文摘要
紧致流形上的加权经验测度在重要性抽样、粒子近似、后验总结、求积和表示学习中具有重要意义。标准的仅权重总结，如普通有效样本量，忽略了支撑集的几何结构。我们引入热核熵剖面，它通过内在热流扩散加权原子并跟踪不同尺度上的不均匀性。对于二阶Rényi熵，该剖面可从成对热核重叠计算得出，并产生几何有效样本量，在匹配分离良好粒子的普通有效样本量时，对附近或重复粒子进行折扣。我们证明了无边界紧致流形的单调性、小尺度和大尺度渐近性、确定性权重一致性以及有界比率自归一化重要性抽样扩展。在球体上，未对数化的剖面分解为球谐能量，可恢复平均方向、冯·米塞斯-费舍尔型和宾汉型总结。基于球体的实验表明，该剖面揭示了仅权重和一阶矩球体总结所遗漏的对映、环带、多模态和重复粒子结构。
英文摘要
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.
