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.

翻译：成分数据——描述相对比例的向量——广泛存在于生态学、地球化学和基因组学等科学领域。这些数据的特征往往具有已知的层级结构（如分类学、系统发育、本体论），然而现有方法要么忽略该结构，要么舍弃了Aitchison几何的内在属性，要么仅针对二叉树设计，要么产生不完整的坐标系。我们提出了PolyILR方法，这是一种与任意树形拓扑对齐的Aitchison切空间的标准正交分解。该构造为每个内部节点定义了捕获完整分支结构的加权局部几何，进而将这些局部几何提升为全局标准正交基，其中每个坐标对应树中的特定位置。在微生物组和单细胞基准测试中，PolyILR提供了稳定且可解释的特征，并支持多尺度树分辨率的推理。此外，我们建立了与softmax分类器的新理论联系，暗示了其在概率建模中的潜在应用。