Bridging Arbitrary and Tree Metrics via Differentiable Gromov Hyperbolicity

TL;DR

This paper introduces DeltaZero, a differentiable optimization framework that effectively approximates arbitrary metric spaces by tree metrics, achieving state-of-the-art distortion with theoretical guarantees.

Contribution

The authors develop a novel differentiable approach to approximate arbitrary metrics by tree metrics, improving upon existing heuristics with better guarantees and statistical justification.

Findings

Achieves state-of-the-art distortion on synthetic datasets.

Provides a differentiable framework enabling gradient-based optimization.

Offers theoretical guarantees better than previous methods.

Abstract

Trees and the associated shortest-path tree metrics provide a powerful framework for representing hierarchical and combinatorial structures in data. Given an arbitrary metric space, its deviation from a tree metric can be quantified by Gromov's $\delta$-hyperbolicity. Nonetheless, designing algorithms that bridge an arbitrary metric to its closest tree metric is still a vivid subject of interest, as most common approaches are either heuristical and lack guarantees, or perform moderately well. In this work, we introduce a novel differentiable optimization framework, coined DeltaZero, that solves this problem. Our method leverages a smooth surrogate for Gromov's $\delta$-hyperbolicity which enables a gradient-based optimization, with a tractable complexity. The corresponding optimization procedure is derived from a problem with better worst case guarantees than existing bounds, and is justified statistically. Experiments on synthetic and real-world datasets demonstrate that our method consistently achieves state-of-the-art distortion.