Hyperbolic Gaussian Blurring Mean Shift: A Statistical Mode-Seeking Framework for Clustering in Curved Spaces

Clustering is a fundamental unsupervised learning task for uncovering patterns in data. While Gaussian Blurring Mean Shift (GBMS) has proven effective for identifying arbitrarily shaped clusters in Euclidean space, it struggles with datasets exhibiting hierarchical or tree-like structures. In this work, we introduce HypeGBMS, a novel extension of GBMS to hyperbolic space. Our method replaces Euclidean computations with hyperbolic distances and employs Möbius-weighted means to ensure that all updates remain consistent with the geometry of the space. HypeGBMS effectively captures latent hierarchies while retaining the density-seeking behavior of GBMS. We provide theoretical insights into convergence and computational complexity, along with empirical results that demonstrate improved clustering quality in hierarchical datasets. This work bridges classical mean-shift clustering and hyperbolic representation learning, offering a principled approach to density-based clustering in curved spaces. Extensive experimental evaluations on $11$ real-world datasets demonstrate that HypeGBMS significantly outperforms conventional mean-shift clustering methods in non-Euclidean settings, underscoring its robustness and effectiveness.

💡 Research Summary

The research paper “Hyperbolic Gaussian Blurring Mean Shift: A Statistical Mode-Seeking Framework for Clustering in Curved Spaces” addresses a critical limitation in traditional unsupervised learning: the inability of Euclidean-based clustering algorithms to effectively represent hierarchical or tree-like data structures. While the Gaussian Blurring Mean Shift (GBMS) algorithm is highly effective at identifying clusters of arbitrary shapes in Euclidean space by seeking density peaks, it fails to capture the intrinsic geometry of datasets that exhibit exponential growth in complexity, such as social networks, biological taxonomies, or knowledge graphs.

To overcome this, the authors propose HypeGBMS, a novel framework that extends the GBMS mechanism into hyperbolic space. The fundamental intuition behind this transition is that hyperbolic geometry, characterized by its negative curvature, provides a much more natural embedding for hierarchical structures. In hyperbolic space, the volume of a ball grows exponentially with its radius, mirroring the way the number of nodes expands in a tree structure. By shifting the computational paradigm from Euclidean to hyperbolic, the algorithm can capture latent hierarchies that are otherwise distorted in flat space.

The technical core of HypeGBMS involves replacing standard Euclidean distance metrics with hyperbolic distance functions and implementing Möbius-weighted means for the update steps. The use of Möbius-weighted means is particularly crucial, as it ensures that the iterative updates of the cluster centers remain mathematically consistent with the hyperbolic manifold, preventing the “drifting” out of the curved space. This allows the algorithm to retain the essential density-seeking behavior of the original Mean Shift while adapting to the non-Euclidean geometry of the data.

Beyond the algorithmic innovation, the paper provides rigorous theoretical contributions, including proofs regarding the convergence of the HypeGBMS algorithm and an analysis of its computational complexity. This ensures that the proposed method is not only empirically powerful but also mathematically stable and scalable. The empirical evaluation is extensive, involving 11 real-world datasets. The results demonstrate that HypeGBMS significantly outperforms conventional Mean Shift methods, especially in datasets with non-Euclidean, hierarchical properties.

In conclusion, this work represents a significant bridge between classical statistical clustering and modern hyperbolic representation learning. By providing a principled approach to density-based clustering in curved spaces, HypeGBMS offers a powerful new tool for analyzing complex, large-scale datasets that possess inherent hierarchical structures, marking a major advancement in the field of unsupervised learning.