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This research introduces BFTC, a novel clustering algorithm that leverages Topological Data Analysis to effectively partition complex datasets with nonconvex or intertwined structures. By utilizing multiscale topological signatures derived from local filtrations, the method outperforms traditional Euclidean-based approaches in capturing intrinsic geometric properties of data manifolds.
Topology-Driven Clustering: Enhancing Performance with Betti Number Filtration
arXiv:2505.04346v2 Announce Type: replace 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.
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.
Machine Learning (cs.LG)
arXiv:2505.04346 [cs.LG]
(or arXiv:2505.04346v2 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2505.04346 Focus to learn more arXiv-issued DOI via DataCite
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