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\section{Related Work}\label{sec:related}

\textbf{Compositional hierarchy methods.} ILR transform provides orthonormal coordinates for compositional data \citep{egozcue2003isometric}. {\color{black}PhILR \citep{silverman2017phylogenetic} aligns this transform with phylogenetic trees, the setting it was designed for, where binary topology is the standard convention; applying it to polytomous trees requires arbitrary binary resolution.} UniFrac \citep{lozupone2005unifrac} incorporates phylogenetic information but produces a dissimilarity measure, not a coordinate system. Phylofactorization \citep{washburne2017phylogenetic} and selbal \citep{rivera2018balances} {\color{black}target biomarker discovery and} identify predictive balances via greedy selection, but yield task-specific contrasts rather than a complete basis. Dirichlet-tree models \citep{mao2022dirichlet} use tree structure for clustering but operate probabilistically, not geometrically. PolyILR complements this line of work by providing a complete orthonormal decomposition for \emph{any} tree. {\color{black}Other work compares proportion-based and compositional normalizations~\citep{yerke2024proportion}.}

\textbf{Trees and geometric representations.} A separate line of work studies geometric representations of trees themselves. Hyperbolic embeddings learn representations of hierarchical data in spaces of constant negative curvature \citep{nickel2017poincare, chami2019hyperbolic, sala2018representation}, while tropical geometry and BHV tree space study geodesics and statistics over spaces \emph{of} trees \citep{billera2001geometry, owen2010fast, monod2018tropical}. These embed trees or treat them as data; PolyILR differs in that the tree is a fixed input that structures a decomposition of the data space.