Density estimation in RKHS with application to Korobov spaces in high dimensions

TL;DR

This paper introduces a kernel-based density estimation method in RKHS, specifically applied to Korobov spaces in high dimensions, achieving near-optimal convergence rates with theoretical and numerical validation.

Contribution

The paper develops a new kernel density estimator in RKHS with error analysis and applies it to Korobov spaces, demonstrating dimension-independent convergence rates.

Findings

Achieves near-optimal convergence rates in high-dimensional Korobov spaces

Provides a general error analysis for kernel density estimators in RKHS

Numerical results confirm theoretical predictions

Abstract

A kernel method for estimating a probability density function (pdf) from an i.i.d. sample drawn from such density is presented. Our estimator is a linear combination of kernel functions, the coefficients of which are determined by a linear equation. An error analysis for the mean integrated squared error is established in a general reproducing kernel Hilbert space setting. The theory developed is then applied to estimate pdfs belonging to weighted Korobov spaces, for which a dimension independent convergence rate is established. Under a suitable smoothness assumption, our method attains a rate arbitrarily close to the optimal rate. Numerical results support our theory.