The Annals of Applied Probability

Approximating geodesics via random points

Erik Davis and Sunder Sethuraman

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Abstract

Given a cost functional $F$ on paths $\gamma$ in a domain $D\subset\mathbb{R}^{d}$, in the form $F(\gamma)=\int_{0}^{1}f(\gamma(t),\dot{\gamma}(t))\,dt$, it is of interest to approximate its minimum cost and geodesic paths. Let $X_{1},\ldots,X_{n}$ be points drawn independently from $D$ according to a distribution with a density. Form a random geometric graph on the points where $X_{i}$ and $X_{j}$ are connected when $0<|X_{i}-X_{j}|<\varepsilon$, and the length scale $\varepsilon=\varepsilon_{n}$ vanishes at a suitable rate.

For a general class of functionals $F$, associated to Finsler and other distances on $D$, using a probabilistic form of Gamma convergence, we show that the minimum costs and geodesic paths, with respect to types of approximating discrete cost functionals, built from the random geometric graph, converge almost surely in various senses to those corresponding to the continuum cost $F$, as the number of sample points diverges. In particular, the geodesic path convergence shown appears to be among the first results of its kind.

Article information

Source
Ann. Appl. Probab., Volume 29, Number 3 (2019), 1446-1486.

Dates
Received: December 2017
Revised: April 2018
First available in Project Euclid: 19 February 2019

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1550566835

Digital Object Identifier
doi:10.1214/18-AAP1414

Mathematical Reviews number (MathSciNet)
MR3914549

Subjects
Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]
Secondary: 58E10: Applications to the theory of geodesics (problems in one independent variable) 62-07: Data analysis 49J55: Problems involving randomness [See also 93E20] 49J45: Methods involving semicontinuity and convergence; relaxation 53C22: Geodesics [See also 58E10] 05C82: Small world graphs, complex networks [See also 90Bxx, 91D30]

Keywords
Geodesic shortest path distance consistency random geometric graph Gamma convergence scaling limit Finsler

Citation

Davis, Erik; Sethuraman, Sunder. Approximating geodesics via random points. Ann. Appl. Probab. 29 (2019), no. 3, 1446--1486. doi:10.1214/18-AAP1414. https://projecteuclid.org/euclid.aoap/1550566835


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