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February 2022 Nonhomogeneous Euclidean first-passage percolation and distance learning
Pablo Groisman, Matthieu Jonckheere, Facundo Sapienza
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Bernoulli 28(1): 255-276 (February 2022). DOI: 10.3150/21-BEJ1341

Abstract

Consider an i.i.d. sample from an unknown density function supported on an unknown manifold embedded in a high dimensional Euclidean space. We tackle the problem of learning a distance between points, able to capture both the geometry of the manifold and the underlying density. We define such a sample distance and prove the convergence, as the sample size goes to infinity, to a macroscopic one that we call Fermat distance as it minimizes a path functional, resembling Fermat principle in optics. The proof boils down to the study of geodesics in Euclidean first-passage percolation for nonhomogeneous Poisson point processes.

Acknowledgments

We want to thank Daniel Carando, Gabriel Larotonda, and Chuck Newman for enlightening conversations and the team of Aristas, especially Yamila Barrera and Alfredo Umfurer, for useful discussions and the implementation of Fermat distance related algorithms. We also thank Steven Damelin and Daniel Mckenzie for private communications that helped us to clarify our respective contributions.

Citation

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Pablo Groisman. Matthieu Jonckheere. Facundo Sapienza. "Nonhomogeneous Euclidean first-passage percolation and distance learning." Bernoulli 28 (1) 255 - 276, February 2022. https://doi.org/10.3150/21-BEJ1341

Information

Received: 1 February 2020; Revised: 1 October 2020; Published: February 2022
First available in Project Euclid: 10 November 2021

Digital Object Identifier: 10.3150/21-BEJ1341

Keywords: distance learning , Euclidean first-passage percolation , nonhomogeneous point processes

Rights: Copyright © 2022 ISI/BS

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Vol.28 • No. 1 • February 2022
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