Topology-Driven Clustering: Enhancing Performance with Betti Number Filtration

TL;DR

This paper addresses clustering of datasets with intricate, intertwined shapes by integrating topological data analysis into clustering. It introduces Betti Number Filtration-based Topological Clustering (BFTC), which builds per-point Vietoris–Rips filtrations, computes Betti numbers up to a chosen dimension, and forms Betti sequences to quantify topological similarity among neighbors. By pruning the initial $k$-NN graph with Betti-sequence-based thresholds and constructing a topology-aware weighted graph, BFTC then uses spectral techniques on the Laplacian followed by $k$-means to obtain clusters. Empirical results on synthetic 2D/3D and real-world datasets show that BFTC outperforms several established clustering methods, demonstrating the value of leveraging higher-dimensional topological features in clustering. The approach offers robust performance across noise settings and provides a flexible framework via Betti-dimension selection for improved adaptability to different data topologies.

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.

Figures (9)

• Figure 1: A comparative study of the different clustering algorithms DBSCAN, $K$-means, Spectral, ToMATo with the proposed BFTC, applied on the Linked-Tori dataset. The performance of BFTC is commendable compared with the rest of the clustering methods.
• Figure 2: The box plots corresponding to cosine similarity scores for (a) $\beta_0$ and (b) $\beta_1$ are presented respectively for the Linked Tori dataset.
• Figure 3: Performance of BFTC (ours) for $3$ synthetic $2$D datasets 3MC, Smile1, Cure-t2-4k taken from clustering benchmark dataset.
• Figure 4: The performance metrics RI, ARI, and NMI for (a) Zoo, (b) Ecoli, and (c) Glass datasets, respectively, with varying Betti numbers are presented. The Betti numbers are varied from dimension $0$ to $5$, and performance becomes optimal for certain Betti numbers that differ for every dataset.
• Figure 5: Performance of BFTC for some Synthetic $3$D Topological datasets; 2 Sphere 2 Circle, Torus Sphere Line, 2 Torus 3 lines respectively.

Theorems & Definitions (6)

• Definition 3.1: Simplicial Complex
• Definition 3.2: Vietoris-Rips Complex
• Definition 3.3: Chain Complex
• Definition 3.4: Persistent Homology
• Definition 3.5: Betti Numbers
• Remark 1