Open Access
October 2016 Shortest path through random points
Sung Jin Hwang, Steven B. Damelin, Alfred O. Hero III
Ann. Appl. Probab. 26(5): 2791-2823 (October 2016). DOI: 10.1214/15-AAP1162


Let $(M,g_{1})$ be a complete $d$-dimensional Riemannian manifold for $d>1$. Let $\mathcal{X}_{n}$ be a set of $n$ sample points in $M$ drawn randomly from a smooth Lebesgue density $f$ supported in $M$. Let $x,y$ be two points in $M$. We prove that the normalized length of the power-weighted shortest path between $x,y$ through $\mathcal{X}_{n}$ converges almost surely to a constant multiple of the Riemannian distance between $x,y$ under the metric tensor $g_{p}=f^{2(1-p)/d}g_{1}$, where $p>1$ is the power parameter.


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Sung Jin Hwang. Steven B. Damelin. Alfred O. Hero III. "Shortest path through random points." Ann. Appl. Probab. 26 (5) 2791 - 2823, October 2016.


Received: 1 February 2013; Revised: 1 November 2015; Published: October 2016
First available in Project Euclid: 19 October 2016

zbMATH: 1353.60028
MathSciNet: MR3563194
Digital Object Identifier: 10.1214/15-AAP1162

Primary: 60F15
Secondary: 53B21 , 60C05

Keywords: conformal metric , power-weighted graph , Riemannian geometry , Shortest path

Rights: Copyright © 2016 Institute of Mathematical Statistics

Vol.26 • No. 5 • October 2016
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