# Boundary Estimation from Point Clouds: Algorithms, Guarantees and Applications

arXiv: https://arxiv.org/abs/2111.03217 · HTML: https://mlsanity.com/paper/2111.03217/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 80 references that are in the map.
- First posted on arXiv 5 Nov 2021.

## Builds on (most cited first, 14 of 14)

- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Error estimates for spectral convergence of the graph Laplacian on random geometric graphs towards the Laplace--Beltrami operator](https://mlsanity.com/paper/1801.10108/) (2018)
- [Lipschitz regularity of graph Laplacians on random data clouds](https://mlsanity.com/paper/2007.06679/) (2020)
- [The game theoretic p-Laplacian and semi-supervised learning with few labels](https://mlsanity.com/paper/1711.10144/) (2017)
- [Rates of Convergence for Laplacian Semi-Supervised Learning with Low Labeling Rates](https://mlsanity.com/paper/2006.02765/) (2020)
- [Consistency of Lipschitz learning with infinite unlabeled data and finite labeled data](https://mlsanity.com/paper/1710.10364/) (2017)
- [Solving PDEs on Unknown Manifolds with Machine Learning](https://mlsanity.com/paper/2106.06682/) (2021)
- [Density Level Sets: Asymptotics, Inference, and Visualization](https://mlsanity.com/paper/1504.05438/) (2015)
- [A maximum principle argument for the uniform convergence of graph Laplacian regressors](https://mlsanity.com/paper/1901.10089/) (2019)
- [Depth for curve data and applications](https://mlsanity.com/paper/1901.00180/) (2019)
- [The limit shape of convex hull peeling](https://mlsanity.com/paper/1805.08278/) (2018)
- [A Hamilton-Jacobi equation for the continuum limit of non-dominated sorting](https://mlsanity.com/paper/1302.5828/) (2013)
- [A continuum limit for the PageRank algorithm](https://mlsanity.com/paper/2001.08973/) (2020)
- [Spectral Clustering Revisited: Information Hidden in the Fiedler Vector](https://mlsanity.com/paper/2003.09969/) (2020)

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

- [Hamilton-Jacobi equations on graphs with applications to semi-supervised learning and data depth](https://mlsanity.com/paper/2202.08789/) (2022)
- [Eikonal depth: an optimal control approach to statistical depths](https://mlsanity.com/paper/2201.05274/) (2022)
- [Boundary Detection Algorithm Inspired by Locally Linear Embedding](https://mlsanity.com/paper/2406.18456/) (2024)
- [Gaussian Linear Functional Manifold Method for Massive Point Cloud Data](https://mlsanity.com/paper/2609.05744/) (2026)

## Abstract (arXiv)

We investigate identifying the boundary of a domain from sample points in the domain. We introduce new estimators for the normal vector to the boundary, distance of a point to the boundary, and a test for whether a point lies within a boundary strip. The estimators can be efficiently computed and are more accurate than the ones present in the literature. We provide rigorous error estimates for the estimators. Furthermore we use the detected boundary points to solve boundary-value problems for PDE on point clouds. We prove error estimates for the Laplace and eikonal equations on point clouds. Finally we provide a range of numerical experiments illustrating the performance of our boundary estimators, applications to PDE on point clouds, and tests on image data sets.