Density Estimation on Rectifiable Sets

TL;DR

This paper introduces a modified kernel density estimator that converges on $d$-rectifiable sets, including algebraic varieties, addressing high-dimensional challenges and demonstrating its effectiveness through numerical experiments.

Contribution

It extends kernel density estimation to $d$-rectifiable sets and algebraic varieties, providing convergence guarantees and rates in these settings.

Findings

Convergence of the modified estimator on rectifiable sets

Specific convergence rate for algebraic varieties

Numerical validation on sparse data

Abstract

Kernel density estimation is a popular method for estimating unseen probability distributions. However, the convergence of these classical estimators to the true density slows down in high dimensions. Moreover, they do not define meaningful probability distributions when the intrinsic dimension of data is much smaller than its ambient dimension. We build on previous work on density estimation on manifolds to show that a modified kernel density estimator converges to the true density on $d-$rectifiable sets. As a special case, we consider algebraic varieties and semi-algebraic sets and prove a convergence rate in this setting. We conclude the paper with a numerical experiment illustrating the convergence of this estimator on sparse data.