Towards Stratified Space Learning: Linearly Embedded Graphs

TL;DR

This paper introduces an algorithm for learning the structure and embedding of linearly embedded graphs from point cloud data, combining computational geometry, algebraic topology, and topological data analysis.

Contribution

It presents a novel algorithm for reconstructing stratified spaces, specifically linearly embedded graphs, from sampled data, with proven correctness under certain assumptions.

Findings

Algorithm successfully reconstructs graph structure from data

Implementation in Julia demonstrates practical applicability

Theoretical guarantees validate the method's correctness

Abstract

In this paper, we consider the simplest class of stratified spaces -- linearly embedded graphs. We present an algorithm that learns the abstract structure of an embedded graph and models the specific embedding 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. The algorithm is implemented in the Julia package http://github.com/yossibokor/Skyler.jl , which we used for the numerical simulations in this paper.