Homological Coordinatization

TL;DR

This paper reviews a method for computing homotopy classes of chain maps and applies it to find meaningful topological maps between complexes, extending unsupervised learning in topological data analysis.

Contribution

It introduces a systematic approach for homotopy class computation and applies it to topological data analysis, bridging algebraic topology and machine learning.

Findings

Effective computation of homotopy classes of chain maps.

Application to topologically meaningful mappings in data analysis.

Extension of unsupervised learning methods to simplicial complexes.

Abstract

In this paper, we review a method for computing and parameterizing the set of homotopy classes of chain maps between two chain complexes. This is then applied to finding topologically meaningful maps between simplicial complexes, which in the context of topological data analysis, can be viewed as an extension of conventional unsupervised learning methods to simplicial complexes.