Bridging Arbitrary and Tree Metrics via Differentiable Gromov Hyperbolicity

Pierre Houédry, Nicolas Courty, Florestan Martin-Baillon, Laetitia Chapel, Titouan Vayer

Keywords: tree metric learning, delta-hyperbolicity, geometrical machine learning, differentiation

TL;DR: A new method to compute a Tree metric from an Arbitrary metric, based on a soft and differentiable version of Gromov Hyperbolicity

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.

Primary Area: General machine learning (supervised, unsupervised, online, active, etc.)

Submission Number: 12963