Bridging Arbitrary and Tree Metrics via Differentiable Gromov Hyperbolicity
This work addresses the challenge of representing hierarchical structures in data for applications in machine learning and data analysis, offering a novel method with improved guarantees over existing approaches.
The paper tackles the problem of bridging arbitrary metrics to tree metrics by introducing DeltaZero, a differentiable optimization framework that leverages a smooth surrogate for Gromov's δ-hyperbolicity, achieving state-of-the-art distortion in experiments on synthetic and real-world datasets.
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 $δ$-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 $δ$-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.