Tree-Structured Orthonormal Decomposition of the Aitchison Simplex

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.

There are many scientific datasets that describe how a fixed total quantity, such as a gut microbial community, a single-cell tissue, or an economy, is split among parts. For this type of data, researchers are largely interested in the relative proportions of the parts, rather than the raw counts, since the totals depend on the measurement instrument. The parts are often organized into a tree of categories; for example, bacteria belong to species, which belong to genera and families, and cells belong to types and lineages. Existing tools to analyze such data either ignore this tree structure, force it into a strict yes/no branching structure, or discard some geometric properties of the data. We developed PolyILR, a way to transform this data into representations that follow the natural shape of the tree, no matter how it branches, in a geometrically principled way. Each coordinate corresponds to one biologically meaningful comparison between groups of species or cells. On microbiome and single-cell datasets, PolyILR finds consistent and scientifically recognizable patterns where prior methods produced unstable, hard-to-interpret results. This gives researchers a reliable way to discover biomarkers and to ask which level of biology drives a signal in their data.