Fitting trees to $\ell_1$-hyperbolic distances

TL;DR

This paper introduces a new approach to fit trees to hyperbolic distances by analyzing hyperbolicity vectors and their norms, providing bounds on embedding errors and revealing differences in real-world versus synthetic hierarchical data.

Contribution

It develops an algorithm that bounds tree embedding error using the $ ext{ell}_1$ norm of hyperbolicity, offering a novel theoretical framework and empirical insights into hierarchical data.

Findings

The $ ext{ell}_1$ error is tightly bounded by the hyperbolicity vector's $ ext{ell}_1$ norm.

Standard datasets for hierarchical analysis differ significantly from synthetic tree-like data.

The proposed algorithm outperforms Gromov's classical results in both theory and practice.

Abstract

Building trees to represent or to fit distances is a critical component of phylogenetic analysis, metric embeddings, approximation algorithms, geometric graph neural nets, and the analysis of hierarchical data. Much of the previous algorithmic work, however, has focused on generic metric spaces (i.e., those with no a priori constraints). Leveraging several ideas from the mathematical analysis of hyperbolic geometry and geometric group theory, we study the tree fitting problem as finding the relation between the hyperbolicity (ultrametricity) vector and the error of tree (ultrametric) embedding. That is, we define a vector of hyperbolicity (ultrametric) values over all triples of points and compare the $\ell_p$ norms of this vector with the $\ell_q$ norm of the distortion of the best tree fit to the distances. This formulation allows us to define the average hyperbolicity (ultrametricity) in terms of a normalized $\ell_1$ norm of the hyperbolicity vector. Furthermore, we can interpret the classical tree fitting result of Gromov as a $p = q = \infty$ result. We present an algorithm HCCRootedTreeFit such that the $\ell_1$ error of the output embedding is analytically bounded in terms of the $\ell_1$ norm of the hyperbolicity vector (i.e., $p = q = 1$) and that this result is tight. Furthermore, this algorithm has significantly different theoretical and empirical performance as compared to Gromov's result and related algorithms. Finally, we show using HCCRootedTreeFit and related tree fitting algorithms, that supposedly standard data sets for hierarchical data analysis and geometric graph neural networks have radically different tree fits than those of synthetic, truly tree-like data sets, suggesting that a much more refined analysis of these standard data sets is called for.