Nonhomogeneous Euclidean first-passage percolation and distance learning

TL;DR

This paper introduces a new distance measure for data points on unknown manifolds that captures both geometric and density information, proving its convergence to a Fermat distance as sample size increases.

Contribution

It develops a novel microscopic distance based on Euclidean first-passage percolation for nonhomogeneous processes and proves its convergence to a macroscopic Fermat distance.

Findings

The microscopic distance converges to the Fermat distance with increasing sample size.

The approach effectively captures manifold geometry and density in high-dimensional data.

Theoretical analysis of geodesics in nonhomogeneous Euclidean first-passage percolation.

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 prove the convergence of this microscopic distance, 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.