Density Estimation on Manifolds with Boundary

TL;DR

This paper introduces a novel kernel density estimation method for manifolds with boundary that accurately estimates boundary location and corrects bias without prior boundary knowledge, improving density estimation in complex geometries.

Contribution

The paper develops a boundary-aware density estimator for manifolds that does not require prior boundary information, using new boundary distance and direction statistics.

Findings

Provides a boundary estimation technique for manifolds with boundary.

Develops a boundary correction method for kernel density estimation.

Achieves uniform bias correction across interior and boundary points.

Abstract

Density estimation is a crucial component of many machine learning methods, and manifold learning in particular, where geometry is to be constructed from data alone. A significant practical limitation of the current density estimation literature is that methods have not been developed for manifolds with boundary, except in simple cases of linear manifolds where the location of the boundary is assumed to be known. We overcome this limitation by developing a density estimation method for manifolds with boundary that does not require any prior knowledge of the location of the boundary. To accomplish this we introduce statistics that provably estimate the distance and direction of the boundary, which allows us to apply a cut-and-normalize boundary correction. By combining multiple cut-and-normalize estimators we introduce a consistent kernel density estimator that has uniform bias, at interior and boundary points, on manifolds with boundary.