High-Dimensional Non-Parametric Density Estimation in Mixed Smooth Sobolev Spaces

TL;DR

This paper introduces adaptive hyperbolic cross density estimators for high-dimensional data that achieve better convergence in mixed smooth Sobolev spaces, reducing the curse of dimensionality in density estimation tasks.

Contribution

The paper proposes novel adaptive hyperbolic cross estimators that perform well in mixed smooth Sobolev spaces, addressing computational and convergence challenges in high-dimensional density estimation.

Findings

Estimator avoids curse of dimensionality under Integral Probability Metric

Demonstrates improved convergence properties in high-dimensional settings

Numerical experiments confirm efficiency and practical applicability

Abstract

Density estimation plays a key role in many tasks in machine learning, statistical inference, and visualization. The main bottleneck in high-dimensional density estimation is the prohibitive computational cost and the slow convergence rate. In this paper, we propose novel estimators for high-dimensional non-parametric density estimation called the adaptive hyperbolic cross density estimators, which enjoys nice convergence properties in the mixed smooth Sobolev spaces. As modifications of the usual Sobolev spaces, the mixed smooth Sobolev spaces are more suitable for describing high-dimensional density functions in some applications. We prove that, unlike other existing approaches, the proposed estimator does not suffer the curse of dimensionality under Integral Probability Metric, including H\"older Integral Probability Metric, where Total Variation Metric and Wasserstein Distance are special cases. Applications of the proposed estimators to generative adversarial networks (GANs) and goodness of fit test for high-dimensional data are discussed to illustrate the proposed estimator's good performance in high-dimensional problems. Numerical experiments are conducted and illustrate the efficiency of our proposed method.