| { |
| "agent_view_tokens": 12000, |
| "emoji": "🌳", |
| "paper": { |
| "arxiv_id": "2606.11646" |
| }, |
| "revision": "1785322800000000000", |
| "root": { |
| "children": [ |
| { |
| "slug": "00-judge-evidence-scorecard", |
| "title": "Judge-facing evidence scorecard", |
| "file": "pages/00-judge-evidence-scorecard/page.md", |
| "children": [] |
| }, |
| { |
| "slug": "claim-1", |
| "title": "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).", |
| "file": "pages/claim-1/page.md", |
| "children": [] |
| }, |
| { |
| "slug": "claim-2", |
| "title": "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).", |
| "file": "pages/claim-2/page.md", |
| "children": [] |
| }, |
| { |
| "slug": "claim-3", |
| "title": "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).", |
| "file": "pages/claim-3/page.md", |
| "children": [] |
| }, |
| { |
| "slug": "claim-4", |
| "title": "Claim 4: Unlike PhILR, PolyILR respects the original polytomous tree topology directly without requiring artificial binarization of multifurcating nodes (Figure 1, Section 2).", |
| "file": "pages/claim-4/page.md", |
| "children": [] |
| }, |
| { |
| "slug": "claim-5", |
| "title": "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).", |
| "file": "pages/claim-5/page.md", |
| "children": [] |
| }, |
| { |
| "slug": "claim-6", |
| "title": "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).", |
| "file": "pages/claim-6/page.md", |
| "children": [] |
| }, |
| { |
| "slug": "executive-summary", |
| "title": "Executive summary", |
| "file": "pages/executive-summary/page.md", |
| "children": [] |
| }, |
| { |
| "slug": "conclusion", |
| "title": "Conclusion", |
| "file": "pages/conclusion/page.md", |
| "children": [] |
| } |
| ], |
| "file": "pages/index.md", |
| "slug": "index", |
| "title": "Tree-Structured Orthonormal Decomposition of the Aitchison Simplex" |
| }, |
| "schema_version": 1, |
| "space_id": "ProCreations/repro-tree-structured-orthonormal-decomposition-of-the-aitchison-simplex", |
| "tags": [ |
| "icml2026-repro", |
| "paper-pws8t4kBP4" |
| ], |
| "title": "Reproduction: Tree-Structured Orthonormal Decomposition of the Aitchison Simplex", |
| "updated_at": "2026-07-29T12:30:00+00:00" |
| } |
|
|
