Bridging Arbitrary and Tree Metrics via Differentiable Gromov Hyperbolicity

2025-05-27

AI Summary

This work addresses the problem of optimally approximating arbitrary metric spaces by tree metrics, quantifying their deviation from tree-like structure. To this end, we propose DeltaZero—the first end-to-end differentiable optimization framework for this task. DeltaZero transforms the nonsmooth Gromov δ-hyperbolicity into a theoretically grounded, smooth surrogate objective, thereby surpassing existing relaxation bounds while preserving statistical validity. Our method integrates differentiable geometric optimization, Gromov hyperbolicity modeling, and smooth relaxation techniques, enabling gradient-based optimization. Extensive experiments on both synthetic and real-world datasets demonstrate that DeltaZero significantly reduces metric distortion and achieves state-of-the-art approximation performance.

Technology Category

• Machine Learning: Optimization
• Search and Optimization: Sampling/Simulation-based Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

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.

Problem

• Quantify deviation from tree metrics using Gromov's hyperbolicity
• Bridge arbitrary metrics to closest tree metrics algorithmically
• Optimize tree metric approximation via differentiable DeltaZero framework

Innovation

• Differentiable optimization for Gromov hyperbolicity
• Gradient-based smooth surrogate for tree metrics
• Improved worst-case guarantees and statistical justification