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arXiv · 2505.21073

Bridging Arbitrary and Tree Metrics via Differentiable Gromov Hyperbolicity

Pierre Houedry·Nicolas Courty·Florestan Martin-Baillon·Laetitia Chapel·Titouan Vayer
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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.

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Pierre Houedry, Nicolas Courty, Florestan Martin-Baillon, Laetitia Chapel, Titouan Vayer. 2025-05-27. Bridging Arbitrary and Tree Metrics via Differentiable Gromov Hyperbolicity. https://arxiv.org/abs/2505.21073

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