Flattening Multiparameter Hierarchical Clustering Functors

TL;DR

This paper introduces a functorial flattening procedure for multiparameter hierarchical clustering, combining topological data analysis and category theory, with empirical validation and a Bayesian update algorithm for learning parameters.

Contribution

It presents a novel functorial flattening method for multiparameter hierarchical clustering and a Bayesian algorithm for parameter learning, bridging topology, category theory, and machine learning.

Findings

The flattening procedure is a functor between clustering categories.

Empirical results show the effectiveness of the method.

The Bayesian update algorithm satisfies a consistency property.

Abstract

We bring together topological data analysis, applied category theory, and machine learning to study multiparameter hierarchical clustering. We begin by introducing a procedure for flattening multiparameter hierarchical clusterings. We demonstrate that this procedure is a functor from a category of multiparameter hierarchical partitions to a category of binary integer programs. We also include empirical results demonstrating its effectiveness. Next, we introduce a Bayesian update algorithm for learning clustering parameters from data. We demonstrate that the composition of this algorithm with our flattening procedure satisfies a consistency property.