Nonparametric density models are of great interest in various scientific and engineering disciplines. Classical density kernel methods, while numerically robust and statistically sound in low-dimensional settings, become inadequate even in moderate higher-dimensional settings due to the curse of dimensionality. In this paper, we introduce a new framework called Variance-Reduced Sketching (VRS), specifically designed to estimate multivariable density functions with a reduced curse of dimensionality. Our framework conceptualizes multivariable functions as infinite-size matrices, and facilitates a new sketching technique motivated by numerical linear algebra literature to reduce the variance in density estimation problems. We demonstrate the robust numerical performance of VRS through a series of simulated experiments and real-world data applications. Notably, VRS shows remarkable improvement over existing neural network estimators and classical kernel methods in numerous density models. Additionally, we offer theoretical justifications for VRS to support its ability to deliver nonparametric density estimation with a reduced curse of dimensionality.

翻译：非参数密度模型在众多科学与工程领域具有重要价值。经典密度核方法虽然在低维场景下具备数值稳健性与统计可靠性，但在中等高维场景中仍因维度诅咒问题而显不足。本文提出一种称为方差缩减草图法的新框架，专门用于在缓解维度诅咒的前提下估计多元密度函数。该框架将多元函数概念化为无限维矩阵，并受数值线性代数文献启发开发出新型草图技术，以降低密度估计问题中的方差。通过一系列仿真实验与真实数据应用，我们验证了VRS框架的稳健数值性能。值得注意的是，在众多密度模型中，VRS相较于现有神经网络估计器与经典核方法均展现出显著提升。此外，我们为VRS提供了理论证明，以支持其在缓解维度诅咒前提下实现非参数密度估计的能力。