Adaptive Hyperbolic Kernels: Modulated Embedding in de Branges-Rovnyak Spaces

TL;DR

This paper introduces adaptive hyperbolic kernels within a curvature-aware RKHS framework, enabling task-specific modulation of hyperbolic embeddings to better model hierarchical data in machine learning applications.

Contribution

It proposes a novel adaptive hyperbolic kernel family based on a curvature-aware de Branges-Rovnyak space, enhancing hyperbolic representation flexibility and performance.

Findings

Outperforms existing hyperbolic kernels on benchmarks

Effectively models hierarchical dependencies in data

Demonstrates improved task-specific embedding modulation

Abstract

Hierarchical data pervades diverse machine learning applications, including natural language processing, computer vision, and social network analysis. Hyperbolic space, characterized by its negative curvature, has demonstrated strong potential in such tasks due to its capacity to embed hierarchical structures with minimal distortion. Previous evidence indicates that the hyperbolic representation capacity can be further enhanced through kernel methods. However, existing hyperbolic kernels still suffer from mild geometric distortion or lack adaptability. This paper addresses these issues by introducing a curvature-aware de Branges-Rovnyak space, a reproducing kernel Hilbert space (RKHS) that is isometric to a Poincare ball. We design an adjustable multiplier to select the appropriate RKHS corresponding to the hyperbolic space with any curvature adaptively. Building on this foundation, we further construct a family of adaptive hyperbolic kernels, including the novel adaptive hyperbolic radial kernel, whose learnable parameters modulate hyperbolic features in a task-aware manner. Extensive experiments on visual and language benchmarks demonstrate that our proposed kernels outperform existing hyperbolic kernels in modeling hierarchical dependencies.