Topology-Driven Clustering: Enhancing Performance with Betti Number Filtration

TL;DR

This paper introduces a novel topological clustering algorithm that leverages Betti number filtration and Betti sequences to better identify complex, intertwined data structures, outperforming existing topology-based methods.

Contribution

The work proposes a new clustering approach using Betti number filtration and Betti sequences to improve performance on complex datasets with intricate shapes.

Findings

Demonstrated improved clustering accuracy on synthetic datasets.

Achieved better performance than existing topology-based algorithms on real-world data.

Effectively captures topological features of complex data structures.

Abstract

Clustering aims to form groups of similar data points in an unsupervised regime. Yet, clustering complex datasets containing critically intertwined shapes poses significant challenges. The prevailing clustering algorithms widely depend on evaluating similarity measures based on Euclidean metrics. Exploring topological characteristics to perform clustering of complex datasets inevitably presents a better scope. The topological clustering algorithms predominantly perceive the point set through the lens of Simplicial complexes and Persistent homology. Despite these approaches, the existing topological clustering algorithms cannot somehow fully exploit topological structures and show inconsistent performances on some highly complicated datasets. This work aims to mitigate the limitations by identifying topologically similar neighbors through the Vietoris-Rips complex and Betti number filtration. In addition, we introduce the concept of the Betti sequences to capture flexibly essential features from the topological structures. Our proposed algorithm is adept at clustering complex, intertwined shapes contained in the datasets. We carried out experiments on several synthetic and real-world datasets. Our algorithm demonstrated commendable performances across the datasets compared to some of the well-known topology-based clustering algorithms.