用最小距离法在希尔伯特距离下实现高精度密度估计，支持混合高斯和对数凹密度。
Density estimation for Hellinger via minimum-distance estimators: mixtures of Gaussians, log-concave, and more
- 基于VC维控制相关概念类，构建适用于希尔伯特距离的最小距离估计框架。
- 首次实现近线性时间算法，学习单变量混合对数凹密度与任意方差混合高斯。
- 适用于多种复杂分布类，样本复杂度接近最优，适合高效密度建模场景。
我们研究从n个样本中准确估计概率密度的问题。经典方法通过最小距离估计器在总变差距离下实现，其理论基础是绑定特定概念类（即Yatracos类）的VC维。本文将该方法扩展至希尔伯特距离，关键在于建立与反向数据处理不等式最新结果的联系，使我们仅需绑定相关概念类的VC维即可设计算法。该框架灵活，可兼容原本为总变差距离设计的快速算法。通过改进Acharya等（2017）的方法，我们首次获得近线性时间算法，用于学习包含单变量混合对数凹密度和任意方差混合高斯的分布类，且样本复杂度接近最优。
原文摘要 · Abstract (English)
We study the task of density estimation, where we hope to accurately estimate a probability density from $n$ samples. A textbook method for density estimation in total variation distance is the minimum-distance estimator approach, where we conclude both the algorithm and the analysis merely from bounding the VC dimension of a particular concept class (the so-called Yatracos class). While this technique has originally yielded sharp guarantees primarily for total variation distance, in this work we extend the minimum-distance estimator approach for learning within Hellinger distance. Our main observation is that we may produce an analogous recipe for Hellinger (where we only require bounding the VC dimension of a related concept class) by drawing connections to recent results yielding reverse data processing inequalities. This recipe is flexible enough to accommodate fast algorithms originally designed for total variation distance; by modifying the approach of Acharya et al. (2017) we conclude the first near-linear time algorithm for learning classes including univariate mixtures of log-concave densities and mixtures of Gaussians (with arbitrary variances), with near-optimal sample complexity.
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