# Exploring Singularities in point clouds with the graph Laplacian: An explicit approach

arXiv: https://arxiv.org/abs/2301.00201 · HTML: https://mlsanity.com/paper/2301.00201/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 3 of its 24 references that are in the map.
- First posted on arXiv 31 Dec 2022.

## Builds on (most cited first, 3 of 3)

- [Gradient Descent Provably Optimizes Over-parameterized Neural Networks](https://mlsanity.com/paper/1810.02054/) (2018)
- [Fine-Grained Analysis of Optimization and Generalization for Overparameterized Two-Layer Neural Networks](https://mlsanity.com/paper/1901.08584/) (2019)
- [Approximation of BV functions by neural networks: A regularity theory approach](https://mlsanity.com/paper/2012.08291/) (2020)

## Abstract (arXiv)

We develop theory and methods that use the graph Laplacian to analyze the geometry of the underlying manifold of datasets. Our theory provides theoretical guarantees and explicit bounds on the functional forms of the graph Laplacian when it acts on functions defined close to singularities of the underlying manifold. We use these explicit bounds to develop tests for singularities and propose methods that can be used to estimate geometric properties of singularities in the datasets.