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

TL;DR

This paper introduces HypeGBMS, a hyperbolic space extension of Gaussian Blurring Mean Shift, enabling effective density-based clustering of hierarchical and curved data structures with theoretical and empirical validation.

Contribution

It presents a novel hyperbolic mean-shift clustering algorithm that captures hierarchical structures, with theoretical analysis and superior performance on real-world datasets.

Findings

HypeGBMS outperforms Euclidean methods on hierarchical datasets.

The method effectively captures latent hierarchies.

Theoretical analysis confirms convergence and complexity.

Abstract

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 M\"obius-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.