Consistent Spectral Clustering under Hyperbolic Geometry

TMLR Paper7131 Authors

Abstract: Spectral clustering is a widely used unsupervised learning method that partitions data by analyzing the spectrum of a similarity graph, where the classical formulations implicitly assume Euclidean geometry. But this assumption becomes inadequate when data exhibit a hierarchical or tree-like structure. In such settings, Euclidean distances distort geodesic relationships, leading to unstable spectral embeddings and degraded clustering performance. Motivated by this limitation, we study spectral clustering under hyperbolic geometry, a natural model for hierarchical data, and propose an intrinsically hyperbolic spectral clustering framework in which the similarity operator is defined using hyperbolic distances after estimating a latent hierarchical root. This construction yields a hyperbolic graph Laplacian whose spectrum better reflects the underlying geometry of the data. We provide a rigorous theoretical analysis establishing the weak consistency of the proposed method under a hyperbolic latent variable model, with convergence rates at least as fast as the classical spectral clustering in Euclidean space. Empirical results on real-world hierarchical datasets demonstrate improved robustness to curvature and hierarchy depth relative to other existing deep and hierarchical clustering benchmarks, highlighting the importance of geometric modeling in spectral methods and positioning hyperbolic geometry as a principled foundation for clustering complex structured data.

Submission Type: Long submission (more than 12 pages of main content)

Previous TMLR Submission Url: https://openreview.net/forum?id=MAKrOcrZKE&nesting=2&sort=date-desc

Changes Since Last Submission: The last submission was desk-rejected because of "modified template", since one of the packages in the TeX file completely modified the template, which was unintentional. We have now identified the package and removed it. We are now submitting the same work in the original template.

Assigned Action Editor: ~Uri_Shaham1

Submission Number: 7131