将树拟合到 $\ell_1$-双曲距离
Fitting trees to $\ell_1$-hyperbolic distances
AI总结：
本文研究将树拟合到距离的问题，通过定义双曲性向量的 $\ell_1$ 范数并提出算法 HCCRootedTreeFit，给出了树嵌入误差的紧致解析界，并揭示标准层次数据集与真正树状数据集的树拟合存在根本差异。
摘要：
构建树来表示或拟合距离是系统发育分析、度量嵌入、近似算法、几何图神经网络以及层次数据分析中的关键组成部分。然而，以往的大部分算法工作都集中在通用度量空间（即没有先验约束的空间）上。利用双曲几何和几何群论数学分析中的若干思想，我们将树拟合问题研究为寻找双曲性（超度量性）向量与树（超度量）嵌入误差之间的关系。也就是说，我们在所有三点组上定义一个双曲性（超度量性）值向量，并将该向量的 $\ell_p$ 范数与最佳树拟合到距离的畸变的 $\ell_q$ 范数进行比较。这种表述使我们能够用双曲性向量的归一化 $\ell_1$ 范数来定义平均双曲性（超度量性）。此外，我们可以将 Gromov 的经典树拟合结果解释为 $p = q = \infty$ 的结果。我们提出了一个算法 HCCRootedTreeFit，使得输出嵌入的 $\ell_1$ 误差在双曲性向量的 $\ell_1$ 范数（即 $p = q = 1$）方面有解析上界，并且这个结果是紧的。此外，与 Gromov 的结果及相关算法相比，该算法在理论和实证性能上有显著不同。最后，我们使用 HCCRootedTreeFit 及相关的树拟合算法表明，据称用于层次数据分析和几何图神经网络的标准数据集，其树拟合与合成的、真正树状数据集的树拟合存在根本差异，这表明需要对这些标准数据集进行更为精细的分析。
英文摘要：
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.
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