Nonparametric models are of great interest in various scientific and
engineering disciplines. Classical kernel methods, while numerically robust and
statistically sound in low-dimensional settings, become inadequate in
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 density functions and nonparametric
regression functions in higher dimensions 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 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 estimation and nonparametric
regression models. Additionally, we offer theoretical justifications for VRS to
support its ability to deliver nonparametric estimation with a reduced curse of
dimensionality.

通过引入一种名为 Variance-Reduced Sketching (VRS) 的新框架，我们特别设计了一种在高维度中估计密度函数和非参数回归函数的方法，其降低了 “维数灾难” 的影响，并通过一系列模拟实验和实际数据应用展示了其在密度估计和非参数回归模型中相较于现有的神经网络估计器和经典核方法的显著改进。此外，我们为 VRS 提供了理论上的证明，支持其在降低维度灾难的同时提供非参数估计能力。