Geometric Inference on Kernel Density Estimates

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.

To appear in SoCG 2015. 36 pages, 5 figures

References in corpus (5)

• Statistical topological data analysis using persistence landscapes
• Confidence sets for persistence diagrams
• Robust Topological Inference: Distance To a Measure and Kernel Distance
• A Gentle Introduction to the Kernel Distance
• Topological consistency via kernel estimation

Cited by in corpus (5)

• Robust Topological Inference: Distance To a Measure and Kernel Distance
• The persistence landscape and some of its properties
• Topological consistency via kernel estimation
• Noise robustness of persistent homology on greyscale images, across filtrations and signatures
• Convergence between Categorical Representations of Reeb Space and Mapper