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Final Published version
Affiliation
Univ Arizona, Dept MathIssue Date
2019-06Keywords
Geodesicshortest path
distance
consistency
random geometric graph
Gamma convergence
scaling limit
Finsler
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INST MATHEMATICAL STATISTICSCitation
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/1550566835Journal
ANNALS OF APPLIED PROBABILITYRights
© Institute of Mathematical Statistics, 2019.Collection Information
This item from the UA Faculty Publications collection is made available by the University of Arizona with support from the University of Arizona Libraries. If you have questions, please contact us at repository@u.library.arizona.edu.Abstract
Given a cost functional F on paths gamma in a domain D subset of R-d, in the form 1 F(gamma) = integral(1)(0) f (gamma(t), gamma(t)) dt , it is of interest to approximate its minimum cost and geodesic paths. Let X-1,...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 < vertical bar X-i - X-j vertical bar < epsilon, and the length scale epsilon = epsilon(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.ISSN
1050-5164Version
Final published versionSponsors
ARO [W911NF-18-1-0311]; Simons sabbatical grantAdditional Links
https://projecteuclid.org/euclid.aoap/1550566835ae974a485f413a2113503eed53cd6c53
10.1214/18-AAP1414