# Tree-Structured Orthonormal Decomposition of the Aitchison Simplex

arXiv: https://arxiv.org/abs/2606.11646 · HTML: https://mlsanity.com/paper/2606.11646/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 56 references that are in the map.
- First posted on arXiv 10 Jun 2026.

## Builds on (most cited first, 5 of 5)

- [Poincar'e Embeddings for Learning Hierarchical Representations](https://mlsanity.com/paper/1705.08039/) (2017)
- [Hyperbolic Graph Convolutional Neural Networks](https://mlsanity.com/paper/1910.12933/) (2019)
- [Dirichlet-tree multinomial mixtures for clustering microbiome compositions](https://mlsanity.com/paper/2008.00400/) (2020)
- [From Logits to Hierarchies: Hierarchical Clustering made Simple](https://mlsanity.com/paper/2410.07858/) (2024)
- [Hypernym Bias: Unraveling Deep Classifier Training Dynamics through the Lens of Class Hierarchy](https://mlsanity.com/paper/2502.12125/) (2025)

## Built on by (most cited first, 1 of 1)

- [Which Question Is Your Attention Metric Answering? Attention Rows as Compositional Data](https://mlsanity.com/paper/2608.14712/) (2026)

## Abstract (arXiv)

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.