{"id":"fd66f017-1950-4dec-bdcd-b891e92ee0a9","arxiv_id":"2606.11646","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"PolyILR supplies a canonical orthonormal decomposition of the Aitchison tangent space aligned with arbitrary tree structures.","lead":"PolyILR constructs an orthonormal basis for compositional data that respects any given tree topology such as taxonomies or phylogenies. A smart generalist might read it to see a geometry-preserving way to extract multiscale features from proportion data in biology and ecology.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.3","headline":"Lifting per-node weighted local bases to one global orthonormal frame may not preserve the Aitchison inner product for nodes with branching factor >2","rationale":"The reader's weakest assumption directly identifies the same lifting step. Because the manuscript text was not supplied, the concern remains a verification gap rather than a demonstrated flaw; the proposed check is a minimal, self-contained test that would confirm or refute the preservation property.","tokens_in":1655,"tokens_out":309,"duration_ms":18059,"concrete_test":"Take a 4-leaf ternary tree (one internal node with three children). Explicitly construct the claimed local weighted basis at that node, apply the lifting map to obtain global clr coordinates, and compute the Gram matrix under the Aitchison inner product; the matrix must be the identity for the claim to hold.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central construction defines a weighted local geometry at each internal node that is claimed to capture the full branching structure, then asserts these can be lifted into a single global orthonormal basis of the Aitchison tangent space. For the claim to hold, the lifted vectors must remain orthogonal under the global Aitchison inner product and span the full (D-1)-dimensional space for any tree topology. Nothing in the abstract rules out the possibility that the lifting step introduces cross terms or rank deficiency precisely when an internal node has three or more children; that is the step whose correctness is least secured by the given description.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper introduces PolyILR as a canonical orthonormal decomposition of the Aitchison tangent space aligned with arbitrary tree topologies for compositional data. It defines a weighted local geometry at each internal node that captures the full branching structure and lifts these local bases to a single global orthonormal basis in which each coordinate corresponds to a specific tree location. The work reports stable interpretable features on microbiome and single-cell benchmarks, multiscale inference capability, and a theoretical link to softmax classifiers.","tokens_in":1769,"tokens_out":437,"duration_ms":16515,"significance":"If the lifting step is shown to preserve the Aitchison inner product and produce a complete basis for any tree (including nodes with branching factor >2), the result would supply a principled, tree-aligned coordinate system for compositional data that respects the intrinsic geometry while enabling hierarchical analysis; this would be a useful contribution to compositional data methods in ecology, genomics, and related fields.","major_comments":[{"comment":"The central construction asserts that weighted local geometries defined at internal nodes can be lifted to a global orthonormal basis of the (D-1)-dimensional Aitchison tangent space while preserving the inner product for arbitrary (including non-binary) trees. The manuscript must supply an explicit verification or inductive argument showing that the lifted vectors remain orthogonal and span the full space when an internal node has three or more children; without this, the claim that the decomposition is canonical and complete for general tree topologies is not secured.","section":"Main construction (lifting step)"}],"minor_comments":[{"comment":"The abstract states existence of the construction and benchmark outcomes but supplies no derivation outline, error analysis, or data-exclusion criteria; the full manuscript should include a concise proof sketch or algorithmic pseudocode for the lifting procedure.","section":"Abstract"},{"comment":"Notation for the weighted local inner product at each node and the precise definition of the lifting map should be introduced with explicit equations rather than descriptive text only.","section":"Methods"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the careful reading and for identifying a point where the theoretical justification can be strengthened. We agree that an explicit inductive argument for the lifting step on non-binary trees will improve the manuscript and will add it in revision.","responses":[{"response":"We acknowledge that while the manuscript states the construction holds for arbitrary trees and provides the local weighted geometry and lifting procedure, an explicit inductive verification of orthogonality and completeness for nodes with branching factor greater than two is not supplied in detail. In the revised manuscript we will insert a new subsection (or appendix) containing a short inductive argument: the base case for binary nodes follows directly from the local orthonormal construction; the inductive step for a node with k>2 children shows that the weighted local basis vectors remain mutually orthogonal under the Aitchison inner product, that their lifts are orthogonal to all previously lifted vectors from other subtrees, and that the resulting set spans the full (D-1)-dimensional tangent space. This addition will make the canonicity claim fully rigorous without altering any other results or claims.","revision_made":"yes","referee_comment":"[Main construction (lifting step)] The central construction asserts that weighted local geometries defined at internal nodes can be lifted to a global orthonormal basis of the (D-1)-dimensional Aitchison tangent space while preserving the inner product for arbitrary (including non-binary) trees. The manuscript must supply an explicit verification or inductive argument showing that the lifted vectors remain orthogonal and span the full space when an internal node has three or more children; without this, the claim that the decomposition is canonical and complete for general tree topologies is not secured."}],"tokens_in":1251,"tokens_out":358,"duration_ms":15497,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"PolyILR constructs an orthonormal basis for the Aitchison tangent space that lines up with any tree topology on the parts. This is the main new piece: earlier methods either ignored the tree, stuck to binary splits, or left the basis incomplete.\n\nThe paper sets up weighted local geometries at each internal node to reflect the full branching, then lifts them into one global frame. The microbiome and single-cell experiments show the coordinates stay stable and support inference at multiple tree scales. The softmax connection is a reasonable extra observation that could link to probabilistic models.\n\nThe soft spot is the lifting itself. Nothing in the abstract rules out cross terms or rank loss when an internal node has three or more children, exactly the point the stress-test flags. Without the explicit steps or an inner-product check in the provided description, it is hard to confirm the global vectors remain orthogonal and span the full space. That part carries the most uncertainty.\n\nThe work targets researchers who handle compositional data with known hierarchies, such as taxonomies in genomics or phylogenies in ecology. A reader who needs coordinates that respect both the Aitchison metric and the tree structure would get concrete value from the construction and the reported benchmarks.\n\nIt deserves a serious referee to examine the derivations on the global basis and any supporting error analysis.","headline":"PolyILR gives a tree-aligned orthonormal basis for Aitchison geometry on arbitrary trees, but the lifting step for nodes with branching factor over 2 still needs checking.","tokens_in":2251,"tokens_out":343,"would_cite":false,"duration_ms":27724,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"PolyILR produces a canonical orthonormal decomposition of the Aitchison tangent space aligned with any tree topology.","keywords":["compositional data","Aitchison geometry","tree topology","orthonormal decomposition","microbiome analysis","single-cell data","softmax connection"],"falsifier":"An explicit counterexample tree (binary or non-binary) for which the lifted coordinates fail to remain orthonormal under the Aitchison inner product or fail to span the full tangent space.","tokens_in":2560,"feed_emoji":"🌳","tokens_out":681,"duration_ms":19992,"temperature":0.7,"pith_summary":"The paper presents PolyILR as a method that decomposes compositional data vectors, which represent relative proportions, into a set of coordinates that follow a given tree hierarchy such as taxonomies or phylogenies. It builds this by first defining a weighted local geometry at each internal node to capture the full branching, then combining those into one global orthonormal basis under the Aitchison inner product. Every resulting coordinate maps directly to a specific location in the tree. A reader would care because this handles both the relative nature of the data and the hierarchical structure at once, enabling stable features and multiscale analysis on data like microbiomes or single-cell counts.","feed_headline":"PolyILR aligns orthonormal coordinates with any tree for proportion data","feed_subtitle":"It turns compositional vectors into stable features that respect both relative proportions and hierarchical branching.","key_machinery":"The PolyILR construction, which defines a weighted local geometry at each internal node to capture full branching structure and lifts it to a global orthonormal basis preserving the Aitchison inner product.","core_discovery":"PolyILR yields a canonical orthonormal decomposition of the Aitchison tangent space aligned with any tree topology, producing stable interpretable features that enable inference at multiscale tree resolution. The 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. It also establishes a novel theoretical connection to softmax classifiers.","pith_inferences":["The same lifting procedure might extend to other hierarchical structures such as ontologies or phylogenies outside the tested domains.","The connection to softmax suggests the coordinates could serve as a drop-in replacement for standard inputs in probabilistic models that already use trees.","Because the basis is canonical for any tree, it could support direct comparison of models trained on different but related hierarchies."],"forward_implications":["Every coordinate in the resulting basis corresponds to a specific location on the input tree.","The features remain stable across different tree resolutions for microbiome and single-cell data.","Inference becomes possible at multiple scales of the same tree without discarding geometry or structure.","A direct theoretical link appears between the decomposition and softmax classifiers."],"fun_headline_variants":["PolyILR orthonormalizes Aitchison data with arbitrary tree structure","Any tree topology yields orthonormal coordinates via PolyILR","PolyILR provides global orthonormal basis from local tree geometries","Aitchison simplex decomposed orthonormally along trees with PolyILR"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"A weighted local geometry defined at each internal node can be lifted to a single global orthonormal basis while preserving the Aitchison inner product and capturing the full branching structure for arbitrary trees.","fun_headline_variants_meta":{"raw":{"variants":["PolyILR orthonormalizes Aitchison data with arbitrary tree structure","Any tree topology yields orthonormal coordinates via PolyILR","PolyILR provides global orthonormal basis from local tree geometries","Aitchison simplex decomposed orthonormally along trees with PolyILR"]},"model":"grok-4.3","cost_usd":0.005585,"raw_usage":{"total_tokens":2653,"prompt_tokens":623,"num_sources_used":0,"completion_tokens":69,"cost_in_usd_ticks":55849500,"prompt_tokens_details":{"text_tokens":623,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1961,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":623,"tokens_out":69,"duration_ms":17973,"temperature":1.0,"reasoning_tokens":1961,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-27T11:05:02.168747+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An explicit counterexample tree (binary or non-binary) for which the lifted coordinates fail to remain orthonormal under the Aitchison inner product or fail to span the full tangent space.","supporting_citations":[],"review_version":1}