Tree-Structured Orthonormal Decomposition of the Aitchison Simplex
This work addresses the need for interpretable, geometry-preserving coordinate systems for compositional data with hierarchical structure, benefiting fields like ecology and genomics.
PolyILR provides a canonical orthonormal decomposition of the Aitchison simplex aligned with any tree topology, yielding stable, interpretable features for compositional data. On microbiome and single-cell benchmarks, it enables multiscale inference and shows a theoretical connection to softmax classifiers.
Compositional data -- vectors encoding relative proportions -- arise across scientific domains, including ecology, geochemistry, and genomics. The features in these data often come with known hierarchical structure (e.g., taxonomies, phylogenies, ontologies), yet existing methods either ignore this structure, discard the intrinsic Aitchison geometry, are designed for binary trees, or yield incomplete coordinate systems. We describe PolyILR, a canonical orthonormal decomposition of the Aitchison tangent space aligned with any tree topology. Our construction defines a weighted local geometry at each internal node capturing full branching structure, then lifts these to a global orthonormal basis where every coordinate corresponds to a specific tree location. On microbiome and single-cell benchmarks, PolyILR yields stable, interpretable features and enables inference at multiscale tree resolution. We also establish a novel theoretical connection to softmax classifiers, suggesting possible applications to probabilistic modeling.