Seeing the forest for the trees

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. We summarize several recent efforts to fit data to trees, including several works of ours that leverage ideas from the mathematical analysis of hyperbolic geometry to study the tree fitting problem as finding the relation between the hyperbolicity (ultrametricity) vector and the error of tree (ultrametric) embedding. Then, we discuss the use (misuse and abuse) of such embeddings in geometric graph neural nets. Joint work with Rishi Sonthalia, Joon-Hyeok Yim, and Isay Katsman.