Geometric Inference on Kernel Density Estimates

TL;DR

This paper demonstrates how geometric inference of point clouds can be effectively performed using kernel density estimates with Gaussian kernels, providing robustness to noise and sampling issues, and introduces algorithms for topological estimation.

Contribution

It introduces a novel approach linking kernel density estimates to geometric inference, with new stability properties and algorithms for topological reconstruction.

Findings

Kernel distance stability results established.

Topological reconstruction from kernel estimates demonstrated.

Algorithm for topology estimation using weighted Vietoris-Rips complexes provided.

Abstract

We show that geometric inference of a point cloud can be calculated by examining its kernel density estimate with a Gaussian kernel. This allows one to consider kernel density estimates, which are robust to spatial noise, subsampling, and approximate computation in comparison to raw point sets. This is achieved by examining the sublevel sets of the kernel distance, which isomorphically map to superlevel sets of the kernel density estimate. We prove new properties about the kernel distance, demonstrating stability results and allowing it to inherit reconstruction results from recent advances in distance-based topological reconstruction. Moreover, we provide an algorithm to estimate its topology using weighted Vietoris-Rips complexes.