Topology-Driven Clustering: Enhancing Performance with Betti Number Filtration

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.

Hyperbolic Fuzzy $C$-Means with Adaptive Weight-based Filtering for Clustering in Non-Euclidean Spaces

Clustering algorithms play a pivotal role in unsupervised learning by identifying and grouping similar objects based on shared characteristics. While traditional clustering techniques, such as hard and fuzzy center-based clustering, have been widely used, they struggle with complex, high-dimensional, and non-Euclidean datasets. In particular, the Fuzzy $C$-Means (FCM) algorithm, despite its efficiency and popularity, exhibits notable limitations in non-Euclidean spaces. Euclidean spaces assume linear separability and uniform distance scaling, limiting their effectiveness in capturing complex, hierarchical, or non-Euclidean structures in fuzzy clustering. To overcome these challenges, we introduce Filtration-based Hyperbolic Fuzzy $C$-Means (HypeFCM), a novel clustering algorithm tailored for better representation of data relationships in non-Euclidean spaces. HypeFCM integrates the principles of fuzzy clustering with hyperbolic geometry and employs a weight-based filtering mechanism to improve performance. The algorithm initializes weights using a Dirichlet distribution and iteratively refines cluster centroids and membership assignments based on a hyperbolic metric in the Poincar\'e Disc model. Extensive experimental evaluations demonstrate that HypeFCM significantly outperforms conventional fuzzy clustering methods in non-Euclidean settings, underscoring its robustness and effectiveness.