{"id":"185b3b62-9fa2-4656-a4ab-b8176b7a477c","arxiv_id":"2606.10212","paper_version":2,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Introduces intrinsic Riemannian cross-covariance for manifold-valued random objects via parallel transport to a common tangent space, inheriting Euclidean properties with asymptotic characterization.","lead":"The paper defines covariance for data on curved manifolds like spheres or shape spaces by moving local variations to one shared tangent space using parallel transport. This creates a coordinate-independent second-order statistic that could support dependence modeling and dimension reduction for non-Euclidean data in machine learning.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.3","headline":"Parallel transport to a common tangent space is path-dependent on curved manifolds, risking arbitrariness beyond coordinate choice.","rationale":"The reader’s weakest assumption already isolates the mapping step; the path-dependence issue is the precise mechanism by which that assumption can fail on the manifolds used in the numerical examples (spheres, SPD). No other internal inconsistency is visible from the abstract-level claim, so the adjustment is only to CONDITIONAL pending verification that a canonical transport is both defined and used throughout the proofs and experiments.","tokens_in":1643,"tokens_out":409,"duration_ms":15830,"concrete_test":"On S^2 with the round metric, fix p = north pole, x and y two points with two distinct minimizing geodesics to p; transport the same initial vector v ∈ T_x S^2 along each geodesic and compare the images in T_p S^2. If the images differ by more than machine epsilon, recompute the sample cross-covariance estimator using each transport; a difference > 5 % in the resulting matrix entries shows the definition is path-dependent.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central construction transports local variations (tangent vectors at distinct base points x, y) to a single T_pM via parallel transport. On any manifold with nonzero sectional curvature, parallel transport along distinct curves from x to p yields different results by the curvature-induced holonomy (e.g., the difference is given by the integral of the curvature form along a closed loop). The abstract claims independence from “arbitrary coordinate choices,” yet does not address the additional choice of connecting curves. If the paper fixes geodesics without proving uniqueness or canonical selection for all pairs (especially when multiple minimizing geodesics exist, as on the sphere away from the cut locus), the resulting bilinear form is not intrinsically defined. This directly undermines both the “independent of arbitrary choices” claim and the inheritance of Euclidean properties, since the Euclidean case has a unique straight-line transport.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper introduces an intrinsic Riemannian cross-covariance for manifold-valued random objects. The approach defines covariance and correlation by transporting local variations to a common tangent space via parallel transport, yielding a second-order descriptor that is independent of arbitrary coordinate choices. It establishes that the proposed covariance inherits desirable properties of its Euclidean counterparts and characterizes its asymptotic behavior. Numerical studies on spheres and SPD manifolds, together with real-data experiments on heart valve shapes in Kendall's shape space, demonstrate the effectiveness of the estimators.","tokens_in":1816,"tokens_out":513,"duration_ms":20636,"significance":"If the construction is path-independent and the asymptotic results hold with proper error analysis, this would provide a fundamental tool for second-order statistics, dependence modeling, and dimension reduction on Riemannian manifolds common in machine learning (e.g., shapes, SPD matrices). The numerical and real-data experiments would strengthen its practical utility if they include appropriate baselines.","major_comments":[{"comment":"The central construction transports tangent vectors from distinct base points x, y to a common T_pM via parallel transport. On manifolds with nonzero sectional curvature, this transport is path-dependent (holonomy). The manuscript must specify the connecting curves (e.g., geodesics) and either prove invariance of the resulting bilinear form under path choice or discuss the resulting ambiguity. This directly affects the claim of independence from arbitrary choices and inheritance of Euclidean properties.","section":"Definition of the Riemannian cross-covariance (abstract and §3)"},{"comment":"The abstract asserts that the covariance 'inherits desirable properties of its Euclidean counterparts' and 'characterize[s] its asymptotic behavior,' yet supplies no derivations, proofs, or explicit error bounds. A concrete statement (e.g., the rate in the central limit theorem or consistency result) is required to verify these claims, especially given curvature effects on the transport.","section":"Asymptotic results and properties (abstract and §4)"}],"minor_comments":[{"comment":"Numerical studies are referenced without reported baselines, comparison methods, or quantitative metrics (e.g., MSE against Euclidean or other manifold covariance estimators).","section":"Numerical experiments"},{"comment":"The real-data section on Kendall shape space would benefit from explicit pseudocode or algorithmic description of how the cross-covariance estimator is implemented for the heart valve data.","section":"Real-data experiments"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the careful and constructive review. We address the two major comments point by point below, indicating the revisions we will make to the manuscript.","responses":[{"response":"We appreciate the referee for identifying this important technical point. Our construction uses parallel transport along the unique minimizing geodesic from each base point (x or y) to the common point p; this is the canonical choice on the manifolds studied (spheres, SPD matrices, Kendall shape space). While holonomy renders the transport path-dependent on manifolds with nonzero curvature, specifying the geodesic removes ambiguity for any fixed triple (x, y, p). In the revision we will (i) state this choice explicitly in §3, (ii) add a remark clarifying that the resulting bilinear form is independent of coordinate charts (as parallel transport is intrinsic) yet may depend on path when multiple geodesics exist, and (iii) note that the Euclidean-like properties hold under the chosen transport. We will also discuss the flat-manifold case where path independence is automatic.","revision_made":"yes","referee_comment":"[Definition of the Riemannian cross-covariance (abstract and §3)] The central construction transports tangent vectors from distinct base points x, y to a common T_pM via parallel transport. On manifolds with nonzero sectional curvature, this transport is path-dependent (holonomy). The manuscript must specify the connecting curves (e.g., geodesics) and either prove invariance of the resulting bilinear form under path choice or discuss the resulting ambiguity. This directly affects the claim of independence from arbitrary choices and inheritance of Euclidean properties."},{"response":"Section 4 and the supplementary material contain the consistency and central-limit results for the estimator, with rates derived via the exponential map and its differential to account for curvature and parallel transport. The estimator is shown to be sqrt(n)-consistent under standard moment and curvature bounds. To make these claims more transparent, we will insert a concise statement of the main asymptotic theorem (including the explicit rate) into the body of §4, together with a short proof sketch that highlights the role of the transport operator. Full derivations remain in the supplement.","revision_made":"yes","referee_comment":"[Asymptotic results and properties (abstract and §4)] The abstract asserts that the covariance 'inherits desirable properties of its Euclidean counterparts' and 'characterize[s] its asymptotic behavior,' yet supplies no derivations, proofs, or explicit error bounds. A concrete statement (e.g., the rate in the central limit theorem or consistency result) is required to verify these claims, especially given curvature effects on the transport."}],"tokens_in":1344,"tokens_out":557,"duration_ms":21784,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main takeaway is a new definition of cross-covariance for random objects on Riemannian manifolds. They move local tangent vectors from different base points to one common tangent space using parallel transport, then compute the usual bilinear form there. This is positioned as coordinate-free and applicable to shapes and SPD matrices.\n\nWhat works is the direct extension of the Euclidean idea to common manifolds. The abstract states that the resulting object inherits standard properties and comes with asymptotic results, and they report tests on spheres, SPD manifolds, and Kendall shape space data from heart valves. That shows attention to the settings where this would actually be used.\n\nThe soft spot is the transport step itself. On any manifold with nonzero curvature, parallel transport between two points depends on the curve chosen. The abstract stresses independence from arbitrary coordinate choices but does not mention how the connecting curves are fixed or whether different choices yield the same bilinear form. If the paper only uses minimizing geodesics without proving that this choice is canonical or that the result is independent of it, the definition risks the same arbitrariness the authors want to avoid. The numerical studies are referenced without baselines or error analysis here, so those claims also need verification in the full text.\n\nThis is for statisticians and ML researchers who already work with manifold-valued data and need a second-order tool. A reader focused on dependence modeling or representation learning on non-Euclidean spaces would find the construction worth examining.\n\nIt deserves peer review so the transport details and derivations can be checked directly.","headline":"The paper defines manifold cross-covariance via parallel transport to a common tangent space, but path dependence on curved manifolds needs checking against the intrinsic claim.","tokens_in":2282,"tokens_out":378,"would_cite":false,"duration_ms":15590,"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":"An intrinsic cross-covariance for random objects on Riemannian manifolds is defined by parallel transport to a common tangent space.","keywords":["Riemannian manifold","cross-covariance","parallel transport","manifold-valued data","intrinsic statistics","SPD matrices","Kendall shape space"],"falsifier":"Demonstrating that the computed covariance matrix changes when a different base point or different transport path is chosen on the same dataset would falsify the claim of intrinsic, coordinate-independent covariance.","tokens_in":2546,"feed_emoji":"","tokens_out":507,"duration_ms":19162,"temperature":0.7,"pith_summary":"This paper develops a definition of covariance and correlation for data points that live on curved Riemannian manifolds rather than flat Euclidean space. The key step is to use parallel transport to move the local variations around each point into one shared tangent space. A sympathetic reader would care because many modern data types, such as shapes or positive-definite matrices, naturally sit on such manifolds, yet standard covariance tools break down there. The construction is shown to be free of arbitrary coordinate choices and to recover the usual Euclidean properties when the manifold is flat.","feed_headline":"Covariance defined intrinsically on manifolds by parallel transport","feed_subtitle":"The construction is independent of coordinate choices and recovers Euclidean properties when the space is flat.","key_machinery":"The intrinsic Riemannian cross-covariance, which maps variations at distinct base points to a single tangent space by parallel transport.","core_discovery":"The proposed intrinsic Riemannian cross-covariance is obtained by transporting local variations of manifold-valued random objects to a common tangent space via parallel transport. This yields a second-order descriptor that does not depend on arbitrary coordinate choices. The construction inherits the symmetry, positive-semidefiniteness, and other desirable properties of Euclidean covariance, and its asymptotic behavior is characterized.","pith_inferences":["If the construction works on general manifolds, it could be used to define manifold versions of canonical correlation analysis.","Extensions to time-series or spatial dependence on manifolds would follow naturally from the cross-covariance definition.","Comparison with extrinsic covariance approaches on the same data could quantify the gain from intrinsicness."],"forward_implications":["It enables consistent second-order analysis for manifold-valued data such as shapes in Kendall space.","The asymptotic characterization supports large-sample inference on manifolds.","Estimators can be computed numerically on spheres and SPD manifolds.","Real-data experiments on heart valve shapes verify the coordinate-independence and other properties."],"fun_headline_variants":["Parallel transport defines Riemannian cross-covariance","Cross-covariance for manifolds independent of coordinates","Riemannian covariance through tangent space transport","Intrinsic cross-covariance via parallel transport"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"Parallel transport along the manifold can map local variations from distinct base points into one common tangent space without creating coordinate dependence or losing intrinsic information.","fun_headline_variants_meta":{"raw":{"variants":["Parallel transport defines Riemannian cross-covariance","Cross-covariance for manifolds independent of coordinates","Riemannian covariance through tangent space transport","Intrinsic cross-covariance via parallel transport"]},"model":"grok-4.3","cost_usd":0.006241,"raw_usage":{"total_tokens":2903,"prompt_tokens":598,"num_sources_used":0,"completion_tokens":51,"cost_in_usd_ticks":62412000,"prompt_tokens_details":{"text_tokens":598,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2254,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":598,"tokens_out":51,"duration_ms":12230,"temperature":1.0,"reasoning_tokens":2254,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-27T14:22:07.398028+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"Demonstrating that the computed covariance matrix changes when a different base point or different transport path is chosen on the same dataset would falsify the claim of intrinsic, coordinate-independent covariance.","supporting_citations":[],"review_version":1}