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    Approximating geodesics via random points

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    Author
    Davis, Erik
    Sethuraman, Sunder
    Affiliation
    Univ Arizona, Dept Math
    Issue Date
    2019-06
    Keywords
    Geodesic
    shortest path
    distance
    consistency
    random geometric graph
    Gamma convergence
    scaling limit
    Finsler
    
    Metadata
    Show full item record
    Publisher
    INST MATHEMATICAL STATISTICS
    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
    Journal
    ANNALS OF APPLIED PROBABILITY
    Rights
    © 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
    1050-5164
    DOI
    10.1214/18-AAP1414
    Version
    Final published version
    Sponsors
    ARO [W911NF-18-1-0311]; Simons sabbatical grant
    Additional Links
    https://projecteuclid.org/euclid.aoap/1550566835
    ae974a485f413a2113503eed53cd6c53
    10.1214/18-AAP1414
    Scopus Count
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    UA Faculty Publications

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