Point-Level Topological Representation Learning on Point Clouds

Vincent Peter Grande, Michael T Schaub

TL;DR: We use topological data analysis to relate global topology to local point features on point clouds.

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 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.
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.

Lay Summary:

Many datasets take the form of a list of points in space.
Previous work has focussed on describing the "shape" of these so-called point clouds.
We now propose a novel method to describe the relationship between each individual point and the global shape of the point cloud.
To do this, we leverage insights from disciplines of Pure Mathematics called Algebraic Topology and Differential Geometry
Finally, we apply our method to data from Biology and Physics and compare it with other methods.

Link To Code: https://github.com/vincent-grande/topf

Primary Area: General Machine Learning->Unsupervised and Semi-supervised Learning

Keywords: Topological Data Analysis, Differential Geometry, Representation Learning, Hodge Laplacian, Simplicial Complex, Persistent Homology, Point Clouds

Submission Number: 11151