Point-Level Topological Representation Learning on Point Clouds

TL;DR

TOPF introduces Topological Point Features, a framework that converts global topological information from a point cloud into per-point descriptors by projecting persistent-homology generators into the harmonic space of the Hodge Laplacian and aggregating locally. The method couples alpha- and Vietoris–Rips filtrations with harmonic projections, normalizations, and density-aware simplicial weighting to produce a |X|×|F| feature matrix suitable for downstream learning. The authors provide theoretical guarantees (e.g., a TOPF sphere-type theorem) and validate TOPF across synthetic and real data, including a new topological clustering benchmark (TCBS) where TOPF-based spectral clustering often outperforms baselines and matches higher-cost topology-aware methods. They also demonstrate robustness to noise, downsampling, and sampling heterogeneity, and show applicability to high-dimensional and latent spaces (e.g., VAE embeddings). The work advances topology-aware feature extraction for point clouds and enables scalable, interpretable integration with standard ML pipelines.

Abstract

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.

Figures (18)

• Figure 1: Topological point features (topf) on real-world and simulated 3d point clouds. For every point, we highlight the largest corresponding topological feature, where colour stands for different features and saturation for the value of the feature. (Cf. \ref{['fig:QualitativeExperiments']})
• Figure 1: Computing Topological Point Features (topf). Input. A point cloud $X$ in $n$-dimensional space. Step 1. To extract global topological information, the persistent homology is computed on an $\alpha$/vr-filtration. The most significant topological features $\mathcal{F}$ across all specified dimensions are selected. Step 2.$k$-homology generators associated to all features $f_{i,k}\in\mathcal{F}$ are computed. For every feature, a simplicial complex is built at a step of the filtration where $f_{i,k}$ is alive. Step 3. The homology generators are projected to the harmonic space. Step 4. The vectors are normalised to obtain vectors $\mathbf{e_k^i}$ indexed over the $k$-simplices. For every point $x$ and feature $f\in\mathcal{F}$, we compute the mean of the entries of $\mathbf{e_k^i}$ corresponding to simplices containing $x$. The output is a $|X|\times|\mathcal{F}|$ matrix which can be used for downstream ml tasks. Optional. We weigh the simplicial complexes resulting in a topologically more faithful harmonic representative in Step 3.
• Figure 2: ph sketch and topf pipeline applied to nalcn channelosome, a membrane protein Kschonsak:2022.Left: Bars represent life times of features grande2023nonisotropic. Centre left: Steps 1&2a, when computing persistent $1$-homology, three classes are more prominent than the rest. Centre right: Step 2b: The selected homology generators. Right: Step 3: The projections of the generators into harmonic space are now each supported on one of the rings.
• Figure 3: topf on 3d real-world and synthetic point clouds. For every point, we highlight the largest corresponding topological feature, where colour stands for the different features and saturation for the value of the feature. (b): Atoms of nalcn channelosome Kschonsak:2022 display three distinct loops. (c): Points sampled in the state space of a Lorentz attractor. The two features correspond to the two lobes of the attractor. (d): Point cloud spaceship of our newly introduced topological clustering benchmark suite (See \ref{['app:BenchmarkSuite']}). (e): Latent space of a vae trained on image patches showing topological structure (See \ref{['fig:figcow']} for details).
• Figure 4: topf on a high-dimensional point cloud. We used topf on $6500$ points sampled from the $24$-dimensional conformation space of cyclooctane Martin2011. We show the isomap projection from $24$ dimensions. Top: Three features automatically selected by topf. Bottom:topf-Clustering for 3 and 4 clusters. topf can correctly cluster similar points according to their topological function. Four clusters, topf can even identify the anomalous points (blue) violating the manifold structure.

Theorems & Definitions (14)

• Definition 2.1: Simplicial complexes
• Definition 2.2: vr complex
• Definition 2.3: Boundary matrices
• Definition 2.4: Hodge Laplacian
• Theorem 2.5: Hodge Decomposition Lim:2020Schaub:2021Roddenberry:2021
• Definition B.1: $n$-dimensional Delaunay triangulation
• Definition B.2: $\alpha^*$-complex of a point cloud
• Definition B.3: Gabriel Simplex
• Definition B.4: $\alpha$-complex of a point cloud, kerber20133dedelsbrunner2011alpha
• Theorem C.1: Topological Point Features of Spheres