发表机构
School of Mathematics, East China University of Science and Technology; Durham University; The London School of Economics and Political Science（华东理工大学数学学院; 杜伦大学; 伦敦政治经济学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一种摊销框架，通过优化对数得分学习核密度估计的带宽映射，其选择器泛化性强且性能优于经典方法，可直接应用于未知密度的有限样本。
AI 中文摘要
核密度估计将有限样本转换为概率密度，但其性能关键依赖于带宽选择。经典选择器通过解析或渐近方式规定样本-带宽规则，或为每个样本求解新的优化问题。本文提出一种摊销框架，该框架通过优化对数得分，在一系列密度估计任务的分布中学习此映射。截断并归一化的有界支持公式可实现异构任务间的稳定学习，而仿射标准化使在单个参考区间上训练的选择器可跨有界区间迁移。在高斯采样、多基准族及随机高斯混合训练下的实验表明，该摊销选择器始终显著优于Silverman规则、Sheather-Jones选择器及最小二乘交叉验证，尤其在小样本和异构样本上增益显著。有限高斯混合凭借其L1逼近特性提供了通用训练机制，以此方式训练的选择器可在不同密度结构间实现强泛化，使同一训练后的选择器可直接应用于未知密度的有限样本，无需指定或拟合分布族。这种广泛适用性与强经验性能的结合，使该框架适用于需将有限样本或集合转换为连续概率密度的众多应用场景。
英文摘要
Kernel density estimation converts finite samples into probability densities, but its performance depends critically on bandwidth selection. Classical selectors prescribe the sample-to-bandwidth rule analytically or asymptotically, or solve a new optimization for each sample. An amortized framework is proposed that instead learns this mapping across a distribution of density-estimation tasks by optimizing the logarithmic score. A truncated-and-renormalized bounded-support formulation enables stable learning across heterogeneous tasks, while affine standardization allows a selector trained on a single reference interval to transfer across bounded intervals. Experiments under Gaussian sampling, a multi-family benchmark, and randomized Gaussian-mixture training show that the amortized selector consistently and substantially outperforms Silverman's rule, the Sheather--Jones selector, and least-squares cross-validation, with especially large gains in small and heterogeneous samples. Finite Gaussian mixtures provide a generic training mechanism supported by their $L^1$ approximation property. Selectors trained in this way generalize strongly across different density structures, allowing the same trained selector to be applied directly to finite samples from unknown densities without specifying or fitting a distributional family. This combination of broad applicability and strong empirical performance makes the framework attractive for a wide range of applications in which finite samples or ensembles must be converted into continuous probability densities.
Comments15 pages, 4 figures, 2 tables, plus 12 pages of supplementary material. Revised author affiliations and acknowledgments; main results unchanged
