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arXiv · 2505.04346

Topology-Driven Clustering: Enhancing Performance with Betti Number Filtration

Arghya Pratihar·Kushal Bose·Swagatam Das
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Abstract

Clustering aims at partitioning data points into groups of similar objects without knowing about the class labels. However, clustering datasets with complex geometric structures, such as nonconvex shapes, multiple scales, or intertwined manifolds, remains challenging for traditional algorithms that primarily rely on Euclidean or kernel-based similarity measures. Topological Data Analysis (TDA), particularly persistent homology, provides a powerful framework for capturing intrinsic structural properties of data, including connected components, loops, and higher-dimensional features across multiple scales. In this work, we propose a novel topological clustering algorithm called \textbf{Betti Number Filtration-based Topological Clustering (BFTC)}. The proposed method constructs local Vietoris-Rips filtrations around each data point and computes Betti numbers up to a prescribed dimension. These Betti numbers across filtration scales form \emph{Betti sequences}, which serve as multiscale topological signatures of local neighborhoods. By comparing Betti sequences among neighboring points, BFTC identifies topologically similar neighbors and refines the neighborhood graph to construct a topology-aware similarity structure and spectral clustering is applied to obtain the final clusters. Experimental results on several synthetic and real-world datasets demonstrate that BFTC effectively clusters complex and intertwined structures and consistently outperforms several state-of-the-art topology-based clustering methods.

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Arghya Pratihar, Kushal Bose, Swagatam Das. 2026-07-21. Topology-Driven Clustering: Enhancing Performance with Betti Number Filtration. https://arxiv.org/abs/2505.04346

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