# 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…

## 23 Citations

Approximating geodesics via random points

- MathematicsThe Annals of Applied Probability
- 2019

Given a `cost' functional $F$ on paths $\gamma$ in a domain $D\subset\mathbb{R}^d$, in the form $F(\gamma) = \int_0^1 f(\gamma(t),\dot\gamma(t))dt$, it is of interest to approximate its minimum cost…

Gromov–Hausdorff limit of Wasserstein spaces on point clouds

- Mathematics
- 2017

We consider a point cloud $X_n := \{ x_1, \dots, x_n \}$ uniformly distributed on the flat torus $\mathbb{T}^d : = \mathbb{R}^d / \mathbb{Z}^d $, and construct a geometric graph on the cloud by…

Exact computation of a manifold metric, via Lipschitz Embeddings and Shortest Paths on a Graph

- Computer Science, MathematicsSODA
- 2020

This paper gives the first exact algorithm for computing a data-sensitive metric called the nearest neighbor metric, and proves the surprising result that a previously published $3-approximation is an exact algorithm.

Exploration of a Graph-based Density-sensitive Metric

- Computer Science
- 2017

This paper shows that the edge-squared metric is equal to previously studied geodesic-based metric defined on points in Euclidean space, and gives fast algorithms to compute sparse spanners of this distance.

Entropic Optimal Transport in Random Graphs

- Computer Science, MathematicsArXiv
- 2022

This paper shows that it is possible to consistently estimate entropic-regularized Optimal Transport (OT) distances between groups of nodes in the latent space, and provides a general stability result forEntropic OT with respect to perturbations of the cost matrix.

Intrinsic Metrics: Exact Equality between a Geodesic Metric and a Graph metric

- Mathematics
- 2017

Some researchers have proposed using non-Euclidean metrics for clustering data points. Generally, the metric should recognize that two points in the same cluster are close, even if their Euclidean…

Convergence rates for estimators of geodesic distances and Frèchet expectations

- Mathematics, Computer ScienceJournal of Applied Probability
- 2018

This paper deals with the study of a natural estimator for the geodesic distance on M, and proves a general convergence result under rather general geometric assumptions on M.

Nonhomogeneous Euclidean first-passage percolation and distance learning

- MathematicsBernoulli
- 2022

Consider an i.i.d. sample from an unknown density function supported on an unknown manifold embedded in a high dimensional Euclidean space. We tackle the problem of learning a distance between…

Shortest path distance in random k-nearest neighbor graphs

- Computer Science, MathematicsICML
- 2012

It is proved that for unweighted kNN graphs, this distance converges to an unpleasant distance function on the underlying space whose properties are detrimental to machine learning.

Analysis of Distance Functions in Graphs

- Computer Science
- 2014

It is shown that in unweighted k-nearest neighbor graphs, the shortest path distance converges to an unpleasant distance function whose properties are detrimental to some machine learning problems.

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