Bridging Arbitrary and Tree Metrics via Differentiable Gromov Hyperbolicity

• URL: http://arxiv.org/abs/2505.21073v2
• Date: Wed, 28 May 2025 09:07:43 GMT
• Title: Bridging Arbitrary and Tree Metrics via Differentiable Gromov Hyperbolicity
• Authors: Pierre Houedry, Nicolas Courty, Florestan Martin-Baillon, Laetitia Chapel, Titouan Vayer,
• Abstract summary: Given an arbitrary metric space, its deviation from a tree metric can be quantified by Gromov's $delta$-hyperbolicity.<n>We introduce a novel differentiable optimization framework, coined DeltaZero, that solves this problem.<n>Our method consistently achieves state-of-the-art distortion.
• Score: 13.751664797063208
• License: http://creativecommons.org/licenses/by/4.0/
• 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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