# Unconstrained and Curvature-Constrained Shortest-Path Distances and their Approximation

arXiv: https://arxiv.org/abs/1706.09441 · HTML: https://mlsanity.com/paper/1706.09441/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Jun 2017.

## Built on by (most cited first, 4 of 4)

- [Non-Parametric Manifold Learning](https://mlsanity.com/paper/2107.08089/) (2021)
- [On the Selection of Tuning Parameters for Patch-Stitching Embedding Methods](https://mlsanity.com/paper/2207.07218/) (2022)
- [Embedding Functional Data: Multidimensional Scaling and Manifold Learning](https://mlsanity.com/paper/2208.14540/) (2022)
- [Semi-supervised Fr'echet Regression](https://mlsanity.com/paper/2404.10444/) (2024)

## Abstract (arXiv)

We study shortest paths and their distances on a subset of a Euclidean space, and their approximation by their equivalents in a neighborhood graph defined on a sample from that subset. In particular, we recover and extend the results of Bernstein et al. (2000). We do the same with curvature-constrained shortest paths and their distances, establishing what we believe are the first approximation bounds for them.