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Geometric Inference on Kernel Density Estimates

Authors Jeff M. Phillips, Bei Wang, Yan Zheng

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Jeff M. Phillips
Bei Wang
Yan Zheng

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Jeff M. Phillips, Bei Wang, and Yan Zheng. Geometric Inference on Kernel Density Estimates. In 31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 34, pp. 857-871, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015) https://doi.org/10.4230/LIPIcs.SOCG.2015.857

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.

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Keywords
  • topological data analysis
  • kernel density estimate
  • kernel distance

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