aiqbench analysis
PolyILR introduces a canonical orthonormal decomposition for compositional data that effectively integrates hierarchical tree structures, addressing limitations in existing methods that ignore topology or restrict themselves to binary trees. Benchmarks on microbiome and single-cell datasets demonstrate its ability to produce stable, interpretable features across multiscale resolutions, while a theoretical link to softmax classifiers suggests broader utility in probabilistic modeling. This advancement offers a robust mathematical framework for handling complex hierarchical data in genomics and ecology.
Tree-Structured Orthonormal Decomposition of the Aitchison Simplex
arXiv:2606.11646v1 Announce Type: new 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.
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.
Accepted at ICML 2026. To appear in PMLR vol. 306
Machine Learning (cs.LG); Quantitative Methods (q-bio.QM); Machine Learning (stat.ML)
arXiv:2606.11646 [cs.LG]
(or arXiv:2606.11646v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2606.11646 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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Full text under the publisher's open licence · 246 words