Shortest Path through Random Points

@article{Hwang2012ShortestPT,
  title={Shortest Path through Random Points},
  author={Sung Jin Hwang and Steven Benjamin Damelin and Alfred O. Hero},
  journal={arXiv: Probability},
  year={2012}
}
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… 

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References

SHOWING 1-10 OF 45 REFERENCES
Limit theory for point processes in manifolds
Let Yi;i ‚ 1; be i.i.d. random variables having values in an m-dimensional manifold M ‰ R d and consider sums P n=1 »(n 1=m Yi;fn 1=m Yig n=1 ), where » is a real valued function deflned on pairs
A Shape Theorem for Riemannian First-Passage Percolation
Riemannian first-passage percolation (FPP) is a continuum model, with a distance function arising from a random Riemannian metric in $\R^d$. Our main result is a shape theorem for this model, which
Asymptotics for weighted minimal spanning trees on random points
For all p[greater-or-equal, slanted]1 let Mp(X1,...,Xn) denote the length of the minimal spanning tree through random variables X1,...,Xn, where the cost of an edge (Xi, Xj) is given by Xi-Xjp. If
Geodesic entropic graphs for dimension and entropy estimation in manifold learning
TLDR
This paper focuses on the geodesic-minimal-spanning-tree (GMST) method, which uses the overall lengths of the MSTs to simultaneously estimate manifold dimension and entropy.
WEIGHTED SUMS OF CERTAIN DEPENDENT RANDOM VARIABLES
1. Let be a probability space and,be an increasing family of sub o'-fields of(we put(c) Let (xn)n=1, 2, •c be a sequence of bounded martingale differences on , that is,xn(ƒÖ) is bounded almost surely
Concentration of measure and isoperimetric inequalities in product spaces
The concentration of measure phenomenon in product spaces roughly states that, if a set A in a product ΩN of probability spaces has measure at least one half, “most” of the points of Ωn are “close”
A note on two problems in connexion with graphs
  • E. Dijkstra
  • Mathematics, Computer Science
    Numerische Mathematik
  • 1959
TLDR
A tree is a graph with one and only one path between every two nodes, where at least one path exists between any two nodes and the length of each branch is given.
Semi-Riemannian Geometry
This chapter develops the basics of differentiable manifolds and semi-Riemannian geometry for the applications in general relativity. It will introduce finitistic substitutes for basic topological
Euclidean models of first-passage percolation
Summary. We introduce a new family of first-passage percolation (FPP) models in the context of Poisson-Voronoi tesselations of ℝd. Compared to standard FPP on ℤd, these models have some technical
Spatial Point Processes and their Applications
TLDR
These lectures will introduce some basic techniques for constructing, manipulating and analysing spatial point patterns.
...
1
2
3
4
5
...