Tree-Structured Orthonormal Decomposition of the Aitchison Simplex

2026-06-10

AI Summary

This work addresses key limitations of existing compositional data analysis methods when applied to hierarchical structures—namely, their neglect of tree topology, violation of Aitchison geometry, restriction to binary trees, or lack of a complete coordinate system. The authors propose PolyILR, the first method to construct an orthonormal basis in the Aitchison simplex tangent space for arbitrary tree topologies. By defining weighted local coordinates at each internal node and lifting them to a global orthonormal basis, PolyILR ensures that each coordinate precisely corresponds to a specific branch of the tree. The approach establishes a theoretical connection to softmax classifiers and enables interpretable, multiscale feature extraction and probabilistic modeling. Experiments on microbiome and single-cell datasets demonstrate that PolyILR yields stable, interpretable features and significantly improves tree-aware inference performance.

Abstract

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.

Problem

• compositional data
• Aitchison geometry
• hierarchical structure
• orthonormal decomposition
• tree topology

Innovation

• PolyILR
• Aitchison geometry
• tree-structured decomposition
• compositional data
• orthonormal basis