# Topological Point Cloud Clustering

arXiv: https://arxiv.org/abs/2303.16716 · HTML: https://mlsanity.com/paper/2303.16716/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 57 references that are in the map.
- First posted on arXiv 29 Mar 2023.

## Builds on (most cited first, 16 of 16)

- [Weisfeiler and Lehman Go Topological: Message Passing Simplicial Networks](https://mlsanity.com/paper/2103.03212/) (2021)
- [Simplicial Neural Networks](https://mlsanity.com/paper/2010.03633/) (2020)
- [Random Walks on Simplicial Complexes and the normalized Hodge 1-Laplacian](https://mlsanity.com/paper/1807.05044/) (2018)
- [Signal Processing on Higher-Order Networks: Livin' on the Edge ... and Beyond](https://mlsanity.com/paper/2101.05510/) (2021)
- [Principled Simplicial Neural Networks for Trajectory Prediction](https://mlsanity.com/paper/2102.10058/) (2021)
- [DTM-based Filtrations](https://mlsanity.com/paper/1811.04757/) (2018)
- [Dist2Cycle: A Simplicial Neural Network for Homology Localization](https://mlsanity.com/paper/2110.15182/) (2021)
- [Using persistent homology and dynamical distances to analyze protein binding](https://mlsanity.com/paper/1412.1394/) (2014)
- [Volume Optimal Cycle: Tightest representative cycle of a generator on persistent homology](https://mlsanity.com/paper/1712.05103/) (2017)
- [The decomposition of the higher-order homology embedding constructed from the k-Laplacian](https://mlsanity.com/paper/2107.10970/) (2021)
- [A Notion of Harmonic Clustering in Simplicial Complexes](https://mlsanity.com/paper/1910.07247/) (2019)
- [Spectral Detection of Simplicial Communities via Hodge Laplacians](https://mlsanity.com/paper/2108.06547/) (2021)
- [Sparse Circular Coordinates via Principal mathbbZ-Bundles](https://mlsanity.com/paper/1809.09269/) (2018)
- [Go with the Flow? A Large-Scale Analysis of Health Care Delivery Networks in the United States Using Hodge Theory](https://mlsanity.com/paper/2110.09637/) (2021)
- [Outlier Detection for Trajectories via Flow-embeddings](https://mlsanity.com/paper/2111.13235/) (2021)
- [Recovering the homology of immersed manifolds](https://mlsanity.com/paper/1912.03033/) (2019)

## Built on by (most cited first, 5 of 5)

- [Disentangling the Spectral Properties of the Hodge Laplacian: Not All Small Eigenvalues Are Equal](https://mlsanity.com/paper/2311.14427/) (2023)
- [Capacity Bounds for Hyperbolic Neural Network Representations of Latent Tree Structures](https://mlsanity.com/paper/2308.09250/) (2023)
- [Point-Level Topological Representation Learning on Point Clouds](https://mlsanity.com/paper/2406.02300/) (2024)
- [A Spectral Framework for Tracking Communities in Evolving Networks](https://mlsanity.com/paper/2412.07378/) (2024)
- [Topology-Driven Clustering: Enhancing Performance with Betti Number Filtration](https://mlsanity.com/paper/2505.04346/) (2025)

## Abstract (arXiv)

We present Topological Point Cloud Clustering (TPCC), a new method to cluster points in an arbitrary point cloud based on their contribution to global topological features. TPCC synthesizes desirable features from spectral clustering and topological data analysis and is based on considering the spectral properties of a simplicial complex associated to the considered point cloud. As it is based on considering sparse eigenvector computations, TPCC is similarly easy to interpret and implement as spectral clustering. However, by focusing not just on a single matrix associated to a graph created from the point cloud data, but on a whole set of Hodge-Laplacians associated to an appropriately constructed simplicial complex, we can leverage a far richer set of topological features to characterize the data points within the point cloud and benefit from the relative robustness of topological techniques against noise. We test the performance of TPCC on both synthetic and real-world data and compare it with classical spectral clustering.