Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Fermat Distance-to-Measure: a robust Fermat-like metric

Jérôme Taupin, Frédéric Chazal

TL;DR

The paper introduces the Fermat Distance-to-Measure (FDTM), a density-free, Fermat-like metric defined for any probability measure by replacing the density term with the Distance-to-Measure (DTM). It establishes foundational geometric and stability properties, proving existence of geodesics and deriving explicit bounds on geodesic length, as well as stability results with respect to perturbations in the measure and the domain. The authors provide a statistically principled estimator based on random samples, with provable convergence rates and a minimax lower bound, and they validate the approach through numerical simulations on the unit circle and comparisons with the classical Fermat distance. This work yields a robust, dimension-agnostic framework for geometry-aware metric learning and clustering for arbitrary measures, with practical implications for tasks where density is ill-defined or multi-scale.

Abstract

Given a probability measure with density, Fermat distances and density-driven metrics are conformal transformations of the Euclidean metric that shrink distances in high density areas and enlarge distances in low density areas. Although they have been widely studied and have shown to be useful in various machine learning tasks, they are limited to measures with density (with respect to Lebesgue measure, or volume form on manifold). In this paper, by replacing the density with the Distance-to-Measure, we introduce a new metric, the Fermat Distance-to-Measure, defined for any probability measure in R^d. We derive strong stability properties for the Fermat Distance-to-Measure with respect to the measure and propose an estimator from random sampling of the same measure, featuring an explicit bound on its convergence rate.

Fermat Distance-to-Measure: a robust Fermat-like metric

TL;DR

The paper introduces the Fermat Distance-to-Measure (FDTM), a density-free, Fermat-like metric defined for any probability measure by replacing the density term with the Distance-to-Measure (DTM). It establishes foundational geometric and stability properties, proving existence of geodesics and deriving explicit bounds on geodesic length, as well as stability results with respect to perturbations in the measure and the domain. The authors provide a statistically principled estimator based on random samples, with provable convergence rates and a minimax lower bound, and they validate the approach through numerical simulations on the unit circle and comparisons with the classical Fermat distance. This work yields a robust, dimension-agnostic framework for geometry-aware metric learning and clustering for arbitrary measures, with practical implications for tasks where density is ill-defined or multi-scale.

Abstract

Given a probability measure with density, Fermat distances and density-driven metrics are conformal transformations of the Euclidean metric that shrink distances in high density areas and enlarge distances in low density areas. Although they have been widely studied and have shown to be useful in various machine learning tasks, they are limited to measures with density (with respect to Lebesgue measure, or volume form on manifold). In this paper, by replacing the density with the Distance-to-Measure, we introduce a new metric, the Fermat Distance-to-Measure, defined for any probability measure in R^d. We derive strong stability properties for the Fermat Distance-to-Measure with respect to the measure and propose an estimator from random sampling of the same measure, featuring an explicit bound on its convergence rate.

Paper Structure

This paper contains 32 sections, 27 theorems, 132 equations, 19 figures, 1 table.

Table of Contents

  1. Introduction
  2. Definitions and Overview of the Results
  3. The Fermat Distance-to-Measure Metric
  4. Overview of the Results
  5. Geodesics and Stability Properties of the FDTM
  6. Existence of Geodesics
  7. Bounding the Length of Geodesics
  8. Case of a Lower Bounded DTM
  9. General Case (sketch of proof)
  10. Stability of the FDTM
  11. Stability with respect to the Measure
  12. Stability with respect to the Domain (sketch of proof)
  13. Estimating the FDTM
  14. Theoretical Convergence Rate
  15. Minimax Lower Bound
  16. ...and 17 more sections

Key Result

Lemma 2

Let $\mu$ be a probability measure over $\mathbb{R}^d$ such that $\mu(\{x\}) < m$ for all $x\in\mathbb{R}^d$. Then $\min d_\mu > 0$.

Figures (19)

  • Figure 1: \ref{['ex:lecam_b']} in the case $m=1$ with the main admissible paths. The dotted line represents the addition of a small density in $\nu$ which allows for a new shorter path (drawn thicker) that is shorter which is the reason for the significant offset in fdtm.
  • Figure 2: $m=0.2$, $\beta=1$
  • Figure 3: $m=0.1$, $\beta=1$
  • Figure 4: $m=0.05$, $\beta=1$
  • Figure 5: $m=0.2$, $\beta=2$
  • ...and 14 more figures

Theorems & Definitions (46)

  • Definition 1: Fermat distance-to-measure (fdtm)
  • Lemma 2
  • Theorem 3: Existence of geodesics
  • Theorem 4: Upper bound on the Euclidean length of geodesics
  • Corollary 5
  • Corollary 6
  • Proposition 7
  • Theorem 8: fdtm stability wrt the measure
  • Theorem 9: fdtm stability wrt the domain
  • Theorem 10: Convergence of the empirical fdtm
  • ...and 36 more