提出新型正交分解方法，可对有层级结构的组成数据进行多尺度分析。
Tree-Structured Orthonormal Decomposition of the Aitchison Simplex
- 基于任意树拓扑构建正交坐标系，保留数据几何结构
- 在微生物组与单细胞数据上实现稳定可解释特征提取
- 适用于需多层级解析的生物数据分析场景
组成数据——编码相对比例的向量——广泛存在于生态学、地球化学和基因组学等领域。这些数据的特征常具有已知的层次结构（如分类体系、系统发育树、本体），但现有方法或忽略此结构，或放弃内在的Aitchison几何，或仅适用于二叉树，或生成不完备的坐标系。本文提出PolyILR，一种与任意树拓扑对齐的Aitchison切空间的标准正交分解。该构造在每个内部节点定义加权局部几何以捕捉完整分支结构，再将其提升为全局正交基，使每个坐标对应树中特定位置。在微生物组与单细胞基准测试中，PolyILR生成稳定且可解释的特征，并支持多尺度树分辨率下的推断。我们还建立了与softmax分类器的新理论联系，暗示其在概率建模中的潜在应用。
原文摘要 · Abstract (English)
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.
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