# Reproduction: Tree-Structured Orthonormal Decomposition of the Aitchison Simplex

## Pages

| Page |
| --- |
| [Judge-facing evidence scorecard](#/00-judge-evidence-scorecard) |
| [Claim 1: Theorem 4.1 proves the constructed matrix V for the PolyILR basis satisfies the contrast property V^T 1 = 0 and orthonormality V^T V = I_{d-1}, making the map φ(x) = V^T log x an isometry (Section 4.2, Theorem 4.1).](#/claim-1) |
| [Claim 2: Proposition 4.2 establishes that PolyILR produces a unique, canonical orthonormal basis for any given tree topology, with the original tree recoverable from the basis's clade support structure (Section 4.3, Proposition 4.2).](#/claim-2) |
| [Claim 3: Algorithm 1 constructs the PolyILR basis by applying weighted Helmert contrasts with Gram-Schmidt orthogonalization at each internal node in depth-first order, then spreading local contrasts to a global leaf-indexed basis by dividing by descendant counts (Section 4.2, Algorithm 1).](#/claim-3) |
| [Claim 4: Unlike PhILR, PolyILR respects the original polytomous tree topology directly without requiring artificial binarization of multifurcating nodes (Figure 1, Section 2).](#/claim-4) |
| [Claim 5: Proposition 7.1 shows the logit-space quotient ℒ is isomorphic to the Aitchison tangent space ℋ, so centered logits equal CLR coordinates (Section 7, Proposition 7.1).](#/claim-5) |
| [Claim 6: The method is validated on the HMP, cMD3, and DISCO microbiome/single-cell datasets, demonstrating stable feature selection and interpretable tree-level (clade) importance aggregation (Section 6, Tables 2-6).](#/claim-6) |
| [Executive summary](#/executive-summary) |
| [Conclusion](#/conclusion) |