Approximating geodesics via random points
dc.contributor.author | Davis, Erik | |
dc.contributor.author | Sethuraman, Sunder | |
dc.date.accessioned | 2019-06-14T22:59:27Z | |
dc.date.available | 2019-06-14T22:59:27Z | |
dc.date.issued | 2019-06 | |
dc.identifier.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 | en_US |
dc.identifier.issn | 1050-5164 | |
dc.identifier.doi | 10.1214/18-AAP1414 | |
dc.identifier.uri | http://hdl.handle.net/10150/632900 | |
dc.description.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. | en_US |
dc.description.sponsorship | ARO [W911NF-18-1-0311]; Simons sabbatical grant | en_US |
dc.language.iso | en | en_US |
dc.publisher | INST MATHEMATICAL STATISTICS | en_US |
dc.relation.url | https://projecteuclid.org/euclid.aoap/1550566835 | en_US |
dc.rights | © Institute of Mathematical Statistics, 2019. | en_US |
dc.rights.uri | http://rightsstatements.org/vocab/InC/1.0/ | |
dc.subject | Geodesic | en_US |
dc.subject | shortest path | en_US |
dc.subject | distance | en_US |
dc.subject | consistency | en_US |
dc.subject | random geometric graph | en_US |
dc.subject | Gamma convergence | en_US |
dc.subject | scaling limit | en_US |
dc.subject | Finsler | en_US |
dc.title | Approximating geodesics via random points | en_US |
dc.type | Article | en_US |
dc.contributor.department | Univ Arizona, Dept Math | en_US |
dc.identifier.journal | ANNALS OF APPLIED PROBABILITY | en_US |
dc.description.collectioninformation | 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. | en_US |
dc.eprint.version | Final published version | en_US |
dc.source.journaltitle | The Annals of Applied Probability | |
dc.source.volume | 29 | |
dc.source.issue | 3 | |
dc.source.beginpage | 1446 | |
dc.source.endpage | 1486 | |
refterms.dateFOA | 2019-06-14T22:59:28Z |