Towards Stratified Space Learning: 2-complexes

TL;DR

This paper introduces an algorithm for learning the structure of 2-complex stratified spaces from point cloud data, combining computational geometry, algebraic topology, and topological data analysis.

Contribution

It presents a novel algorithm that accurately reconstructs the abstract structure of embedded 2-complexes from sampled point clouds, with proven correctness under certain conditions.

Findings

Algorithm successfully reconstructs 2-complex structures from data

Theoretical guarantees ensure correctness under specific assumptions

Integrates tools from geometry, topology, and data analysis

Abstract

In this paper, we consider a simple class of stratified spaces -- 2-complexes. We present an algorithm that learns the abstract structure of an embedded 2-complex from a point cloud sampled from it. We use tools and inspiration from computational geometry, algebraic topology, and topological data analysis and prove the correctness of the identified abstract structure under assumptions on the embedding.