# Point-Level Topological Representation Learning on Point Clouds

arXiv: https://arxiv.org/abs/2406.02300 · HTML: https://mlsanity.com/paper/2406.02300/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 80 references that are in the map.
- First posted on arXiv 4 Jun 2024.

## Builds on (most cited first, 25 of 25)

- [PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation](https://mlsanity.com/paper/1612.00593/) (2016)
- [UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction](https://mlsanity.com/paper/1802.03426/) (2018)
- [ShapeNet: An Information-Rich 3D Model Repository](https://mlsanity.com/paper/1512.03012/) (2015)
- [node2vec: Scalable Feature Learning for Networks](https://mlsanity.com/paper/1607.00653/) (2016)
- [Persistence Images: A Stable Vector Representation of Persistent Homology](https://mlsanity.com/paper/1507.06217/) (2015)
- [An introduction to Topological Data Analysis: fundamental and practical aspects for data scientists](https://mlsanity.com/paper/1710.04019/) (2017)
- [Topological Autoencoders](https://mlsanity.com/paper/1906.00722/) (2019)
- [Ripser: efficient computation of Vietoris-Rips persistence barcodes](https://mlsanity.com/paper/1908.02518/) (2019)
- [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)
- [Persistent Laplacians: properties, algorithms and implications](https://mlsanity.com/paper/2012.02808/) (2020)
- [GRIL: A 2-parameter Persistence Based Vectorization for Machine Learning](https://mlsanity.com/paper/2304.04970/) (2023)
- [The decomposition of the higher-order homology embedding constructed from the k-Laplacian](https://mlsanity.com/paper/2107.10970/) (2021)
- [Graphcode: Learning from multiparameter persistent homology using graph neural networks](https://mlsanity.com/paper/2405.14302/) (2024)
- [A Notion of Harmonic Clustering in Simplicial Complexes](https://mlsanity.com/paper/1910.07247/) (2019)
- [Helmholtzian Eigenmap: Topological feature discovery & edge flow learning from point cloud data](https://mlsanity.com/paper/2103.07626/) (2021)
- [WSDesc: Weakly Supervised 3D Local Descriptor Learning for Point Cloud Registration](https://mlsanity.com/paper/2108.02740/) (2021)
- [Sparse Circular Coordinates via Principal mathbbZ-Bundles](https://mlsanity.com/paper/1809.09269/) (2018)
- [Diffeomorphic interpolation for efficient persistence-based topological optimization](https://mlsanity.com/paper/2405.18820/) (2024)
- [Topological Point Cloud Clustering](https://mlsanity.com/paper/2303.16716/) (2023)
- [Topological Node2vec: Enhanced Graph Embedding via Persistent Homology](https://mlsanity.com/paper/2309.08241/) (2023)
- [Non-isotropic Persistent Homology: Leveraging the Metric Dependency of PH](https://mlsanity.com/paper/2310.16437/) (2023)
- [Circular Coordinates for Density-Robust Analysis](https://mlsanity.com/paper/2301.12742/) (2023)
- [Novel definition and quantitative analysis of branch structure with topological data analysis](https://mlsanity.com/paper/2402.07436/) (2024)

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

- [Topological Trajectory Classification and Landmark Inference on Simplicial Complexes](https://mlsanity.com/paper/2412.03145/) (2024)
- [On the Sheafification of Higher-Order Message Passing](https://mlsanity.com/paper/2509.23020/) (2025)
- [Efficient Sparsification of Simplicial Complexes via Local Densities of States](https://mlsanity.com/paper/2502.07558/) (2025)
- [Selecting Interpretable Circular Coordinates from Data](https://mlsanity.com/paper/2607.08230/) (2026)

## Abstract (arXiv)

Topological Data Analysis (TDA) allows us to extract powerful topological and higher-order information on the global shape of a data set or point cloud. Tools like Persistent Homology or the Euler Transform give a single complex description of the global structure of the point cloud. However, common machine learning applications like classification require point-level information and features to be available. In this paper, we bridge this gap and propose a novel method to extract node-level topological features from complex point clouds using discrete variants of concepts from algebraic topology and differential geometry. We verify the effectiveness of these topological point features (TOPF) on both synthetic and real-world data and study their robustness under noise and heterogeneous sampling.