Tree-Structured Orthonormal Decomposition of the Aitchison Simplex

Area of Science:

• Bioinformatics
• Computational Biology
• Data Science

Background:

• Compositional data, which represent relative proportions, are common in ecology, geochemistry, and genomics.
• Existing methods often fail to leverage the inherent hierarchical structure (e.g., taxonomies, phylogenies) of these data or disregard the Aitchison geometry.
• Current approaches may be limited to binary trees or produce incomplete coordinate systems.

Purpose of the Study:

• To develop a novel method, PolyILR, for analyzing compositional data that respects hierarchical structures.
• To create a canonical orthonormal decomposition of the Aitchison tangent space aligned with arbitrary tree topologies.
• To generate stable and interpretable features from complex biological data.

Main Methods:

• PolyILR decomposes compositional data using a canonical orthonormal transformation aligned with any tree topology.
• It defines a weighted local geometry at each internal node to capture branching structure.
• These local geometries are then integrated into a global orthonormal basis.

Main Results:

• PolyILR yields stable and interpretable features from compositional data.
• The method enables inference across multiple resolutions of the data's tree structure.
• Performance was validated on microbiome and single-cell benchmark datasets.

Conclusions:

• PolyILR effectively handles compositional data with hierarchical structures, offering a more complete representation.
• The method's features are stable and interpretable, facilitating multiscale analysis.
• A theoretical link to softmax classifiers suggests potential in probabilistic modeling.