{"id":"52f01f0f-47e0-4ce2-b6d8-78f55998404f","arxiv_id":"2607.25163","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":1,"one_line_summary":"A parallel-transport-based intrinsic generalization of active subspaces to Riemannian manifolds, with second-order intrinsic/extrinsic equivalence and 2-sphere demonstrations.","lead":"This paper extends active subspaces, an eigenvalue-ordered sensitivity-analysis tool, to functions on curved spaces by parallel-transporting gradients to a central tangent space. It proves the intrinsic version matches the standard embedding-based version to second order in the sampling radius, with exact formulas on spheres relevant to shape analysis.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Ridge-recovery rate in abstract/§2.9 is not entailed by Prop. 3 as written: transferring Euclidean bound (8) to a geodesic ball skips the centering/whitening that (8) requires.","rationale":"The reader's weakest_assumption was the geodesic-ball/injectivity-radius domain restriction, which the paper explicitly scopes and honestly defers. My independent concern is a different, more consequential gap: even inside that local scope, the advertised ridge-recovery rate is not formally derived. Prop. 3 itself appears mathematically sound—the hypersphere case is rigorous and the general-manifold sketch is plausible—so I do not see grounds to reject or to demand a stronger verdict. The reader's CONDITIONAL verdict already flags informal ridge-transfer and missing code/error bars; my concern sharpens the first of those. I therefore recommend keeping CONDITIONAL, with the condition explicitly including a proof of the ridge-bound transfer or a caveat that ridge recovery is only demonstrated numerically.","tokens_in":35476,"tokens_out":24020,"duration_ms":229858,"concrete_test":"On S^2 with p0=(0,0,1) and f(x)=a^T x with a in the xy-plane, take µ uniform on a geodesic ball of radius R. Compute the coordinate-gradient matrix C_coord of g(y)=f(exp_{p0} y) and the conditional-average ridge h(w_1^T y). Compare the L2 error against bound (8) using the trailing eigenvalues of C_coord, first with the raw uniform measure on the ball and then after centering/whitening y in normal coordinates. If the raw version violates (8) or whitening changes the active direction by more than O(R^2), the §2.9 transfer needs an additional theorem; if it holds, the missing normalization is technical and a proof should be supplied.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Prop. 3 establishes O(R^2) agreement between the intrinsic and extrinsic central matrices, and that argument is sound within the stated geodesic-ball scope. The fragile step is the sentence in §2.9: 'The same bound (8) therefore holds on the manifold up to O(R^2) corrections.' Bound (8) is the Euclidean ridge-approximation theorem of §1.5, whose hypotheses include a compact regular-closed domain that has been centered and rescaled so that ∫xρ=0 and ∫xx^Tρ=I_n. Def. 2's µ is an arbitrary wrapped measure on a geodesic ball; no centering or whitening of the normal-coordinate measure is imposed, and whitening would change the coordinate directions whose eigenpairs are being compared. Eigenvalue agreement of two matrices to O(R^2) does not by itself produce the conditional-average h or control the L2 error in (8). The advertised 'curvature-limited quadratic rate' of ridge recovery (abstract; Figs. 6–7) is therefore an extrapolation, not a derived theorem. This does not refute Prop. 3, but it weakens the practical claim that the AMG ordering inherits the Euclidean ridge-certification guarantee.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes an intrinsic generalization of Euclidean active subspaces to scalar functions on Riemannian manifolds. The central construction, active manifold-geodesics (AMG), replaces the Euclidean gradient outer product (4) with a parallel-transported gradient outer product G0 on a central tangent space (Def. 2, Eq. (12)). Its eigenvalues are shown to be exact mean-squared directional derivatives along a transported eigenvector frame (Lemma 1). The paper then contrasts this intrinsic construction with the extrinsic embedding-based average C_ι (Eq. (16)). The main quantitative result is Prop. 3: on a geodesic ball of radius R below the injectivity radius, the centrally projected extrinsic matrix E0^T C_ι E0 and the intrinsic G0 agree to O(R^2), with eigenvalues agreeing to O(R^2) and dominant eigenspaces to O(R^2/η) under a spectral gap η. A closed-form projection-versus-transport identity is given on hyperspheres (Lemma 2). The paper also claims a 'curvature-limited quadratic rate' for ridge recovery in normal coordinates and illustrates the formalism on the 2-sphere, with applications to preshape spaces motivated.","tokens_in":35824,"tokens_out":6458,"duration_ms":60237,"significance":"If the results hold, the paper supplies a principled, coordinate-free extension of active subspaces to manifold-valued parameter domains, retaining the eigenvalue-ordered interpretability of the Euclidean theory. It makes a precise quantitative statement about when the intrinsic and extrinsic perspectives coincide, and it identifies a concrete failure mode of the common 'project-then-order' practice (Remark 7). The paper is largely self-contained: numbered definitions are clear, Lemma 1, Thm. 1, Lemma 2, and the hypersphere case of Prop. 3 are proven in detail, and the explicit hypersphere constant is a genuine strength. The numerical experiments, while simple, do illustrate the mechanisms. The main weaknesses are an unproven transfer of the Euclidean ridge-approximation bound to manifolds (§2.9) and a sketched, not fully tracked, argument for the general-manifold case of Prop. 3.","major_comments":[{"comment":"The claim 'The same bound (8) therefore holds on the manifold up to O(R^2) corrections' is not entailed by the preceding derivation. Prop. 3 establishes O(R^2) agreement between the matrices E0^T C_ι E0 and G0, and the Weyl/Davis–Kahan consequences for eigenvalues and eigenspaces. Bound (8), however, is an L2 approximation statement with h the conditional average over inactive coordinates; its Euclidean proof requires the domain to be centered and whitened (∫ x ρ dx = 0 and ∫ xx^T ρ dx = I_n, §1.5). Def. 2 imposes no such normalization on the wrapped measure µ, and whitening would change the coordinate directions whose eigenpairs are being compared. An O(R^2) eigenvalue agreement does not by itself produce the conditional average h or control the L2 error in (8). Thus the abstract's 'derived ridge recovery at a curvature-limited quadratic rate' is an extrapolation rather than a theorem.","section":"§2.9 and Abstract"},{"comment":"The general-manifold part of Prop. 3 is currently a proof sketch rather than a complete proof. Step 4 states 'Taylor’s theorem with integral remainder gives F(t)[w] = E0w + tII(x̄,w) + O(t^2)' and 'applying Grönwall’s inequality gives V(t) = V(0) + O(t^2)', but the remainders are not tracked and the claimed dependence of the constant C on the second fundamental form and curvature is not demonstrated. Since Prop. 3 is the central theorem, the statement that C depends only on the second fundamental form of ι(M) and its curvature over the support needs to be substantiated with a complete argument, or the theorem should be stated with C as an unspecified geometric constant and the proof supplied in an appendix. The hypersphere case, with the explicit constant C = 1 + R^2/4, is fine.","section":"Prop. 3 / Appendix D, Step 4"}],"minor_comments":[{"comment":"The Euclidean assumption 'without loss of generality ... centering and rescaling ... ∫ x ρ dx = 0 and ∫ xx^T ρ dx = I_n' is not mirrored in Def. 2, where µ is arbitrary. Since this mismatch underlies the ridge-transfer issue, it would help to state explicitly in §2.7 that no centering/whitening is assumed in the manifold construction.","section":"§1.5 / §2.7"},{"comment":"The same symbol C is used both as the generic constant in ||E0^T C_ι E0 - G0||_2 ≤ C ∫ ... and as the specific value C = 1 + R^2/4 on hyperspheres. The notation would be clearer if the constant carried a subscript, e.g., C_R.","section":"Prop. 3 display"},{"comment":"Equation (23) contains a typographical artifact: 'τ 1−n −τ− − − − → τ→0 ∞' should read '→ ∞ as τ → 0'.","section":"Appendix B, Eq. (23)"},{"comment":"Reference [13] has a typo in the title: 'Emergine Ideas' should be 'Emerging Ideas'.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The core construction and the hypersphere results are sound and the paper is within scope for a numerical analysis / scientific computing journal. The main revision should focus on either proving or explicitly qualifying the ridge-recovery claim in §2.9 and tightening the proof of the general-manifold case of Prop. 3. These are fixable within the manuscript's scope; I do not see a need for rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"What you should know first: this is a real contribution, not a repackaging. The parallel-transport-averaged gradient outer product G0 is new, and Lemma 1 gives it an exact mean-squared-directional-derivative reading, the same interpretability the Euclidean theory has. Prop. 3 is the right comparison between intrinsic transport and extrinsic projection, with explicit constants on the sphere, and Thm. 1 correctly diagnoses why projected dominance fails. I agree with the reader's conditional verdict.\n\nThe paper is self-contained and careful. The proofs of Lemma 1, Thm. 1, Lemma 2, and Prop. 3 are coherent; the hypersphere closed-form identity is a genuinely useful piece of geometry. Related work is thorough and fair, and the self-citations are background, not load-bearing.\n\nThe weakest point is the ridge-recovery claim in §2.9 and the abstract. The Euclidean bound (8) assumes a domain that is centered and whitened (zero mean, identity covariance). Def. 2's wrapped measure imposes neither. Eigenvalue agreement to O(R^2) between two matrices does not give you the conditional average h, nor does it control the L2 error in (8). So the \"curvature-limited quadratic rate\" advertised in the abstract and Figs. 6–7 is, at this stage, a numerical observation rather than a derived theorem. The author should either prove a manifold ridge bound under explicit conditions on µ (for instance, normal-coordinate mean zero and identity covariance after choosing a frame) or soften the abstract. This does not invalidate Prop. 3, but it is a load-bearing advertisement.\n\nOther soft spots are minor: the general-manifold constant in Prop. 3 is left qualitative, the O(R^2) frame expansions are sketched rather than fully tracked, and the numerical sections would be stronger with code and repeated-seed error bars on all plots. Fig. 8 has error bars; Figs. 6/7 do not.\n\nWho this is for: anyone doing dimension reduction for scalar quantities of interest on manifolds, especially shape spaces. It deserves a serious referee. I would send it out, and ask the author to fix the ridge-transfer claim and provide code/error bars. The core science is sound.","headline":"A genuinely new intrinsic generalization of active subspaces with a solid core (Lemma 1, Prop. 3), but the advertised ridge-recovery rate goes beyond what the proofs actually establish.","tokens_in":36228,"tokens_out":1954,"would_cite":true,"duration_ms":19288,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62R30","53B21","62H25","65F15"],"pacs":[],"model":"deepseek-v4-flash","headline":"Active subspaces generalize to Riemannian manifolds by parallel-transporting gradients to a central tangent space, where intrinsic and extrinsic views agree to second order.","keywords":["active subspaces","Riemannian manifolds","parallel transport","active manifold-geodesics","dimension reduction","shape analysis","preshape spaces","hyperspheres"],"falsifier":"On the unit 2-sphere, center at the north pole, sample a wrapped uniform measure on a geodesic ball of radius R, and use a function with known gradient, e.g. f(x)=a·x. Numerically evaluate ||E0^T C_ι E0 − G0||_2 for shrinking R and verify it scales as O(R^2) with the explicit constant 1+R^2/4 from Lemma 2 as R approaches the injectivity radius. A rate slower than quadratic or a violation of the stated constant would refute Proposition 3.","tokens_in":1639,"feed_emoji":"🌐","tokens_out":2181,"duration_ms":65754,"temperature":0.7,"pith_summary":"The paper extends active subspaces, the eigenvalue-ordered average of squared directional derivatives that reveals which parameter directions change a scalar output most, from Euclidean domains to functions defined on Riemannian manifolds such as spheres. The key move is to define an intrinsic average tensor by parallel-transporting each Riemannian gradient along the unique geodesic to a central tangent space. The central theorem shows that this intrinsic average and the standard embedding-based gradient average, compressed to the same central tangent space, agree to second order in the geodesic radius of the sampled domain: eigenvalues agree to O(R^2) and dominant eigenspaces to O(R^2/η) under a spectral gap η. This validates active manifold-geodesics as the principled manifold generalization of active subspaces, while warning against extending activity by projecting the dominant ambient eigenvector. Because landmark and elastic-curve preshape spaces are hyperspheres, the construction applies directly to shape analysis.","feed_headline":"Curved active subspaces: two views match to second order","feed_subtitle":"Intrinsic gradient transport and embedding projection align locally, so both recover the same dominant directions on curved domains.","key_machinery":"The central object is the parallel-transport gradient outer product (Definition 2), a (0,2)-tensor on the central tangent space obtained by radial parallel transport of the Riemannian gradient before averaging against a wrapped measure. Its eigendecomposition defines active manifold-geodesics (Definition 3): exponential images of spans of leading eigenvectors. Lemma 1 gives the exact mean-squared-directional-derivative reading of eigenvalues; Lemma 2 gives the closed-form projection-versus-transport identity on hyperspheres, which Proposition 3 integrates into the O(R^2) agreement bound.","core_discovery":"The central claim is Proposition 3: on a Riemannian manifold, for a scalar function sampled on a geodesic ball of radius R below the injectivity radius, the intrinsic parallel-transport gradient average G0 and the extrinsic embedding-based average C_ι restricted to the central tangent space satisfy ||E0^T C_ι E0 − G0||_2 ≤ C R^2 sup ||∇f||^2, with C explicit on hyperspheres. Consequently the two spectra agree to O(R^2) and, under a spectral gap η, the dominant eigenspaces agree to O(R^2/η). Lemma 1 makes the intrinsic eigenvalues exact: each eigenvalue is the mean-squared directional derivative along the eigenvector field obtained by inverse parallel transport. Together these establish activ","pith_inferences":["A practical audit rule follows: estimate the injectivity radius of a data-driven manifold and choose sampling radii R small enough that the O(R^2) term is well below the spectral gap η; otherwise the intrinsic and extrinsic spectra are indistinguishable.","The general-manifold constant in Proposition 3 is left qualitative; the hypersphere identity provides a template for deriving quantitative constants on other symmetric spaces.","The foliation-versus-constant dichotomy suggests a testable criterion: if a function's inactive directions fail to bracket-generate on a connected domain, the data are consistent with a low-dimensional manifold ridge.","A direct numerical check on a nonspherical manifold, such as the SPD manifold, would test whether the O(R^2) scaling holds beyond hyperspheres and whether the eigenspace error scales as O(R^2/η)."],"forward_implications":["On any Riemannian manifold, the eigenvalues of G0 retain the exact interpretation as mean-squared directional derivatives along unit-norm transported fields, so the importance ordering survives on curved domains.","Within a geodesic ball of radius R below the injectivity radius, intrinsic and extrinsic eigenvalue decompositions agree to O(R^2) and dominant eigenspaces to O(R^2/η), so both perspectives are locally consistent for identifying activity.","Extending activity by projecting the dominant ambient eigenvector can fail: wherever that eigenvector lies in the normal space, its projection vanishes and the ordering is conflated; one must instead eigendecompose the centrally projected representation or use intrinsic transport.","Because landmark preshape and length-normalized elastic-curve preshape spaces are hyperspheres, the construction gives a response-driven dimension reduction for functions of shape along preshape geodesics.","When trailing eigenvalues vanish, f is constant along the inactive distribution; if that distribution Lie-generates a foliation, f becomes a manifold ridge function over the leaves, and otherwise it is constant outright."],"fun_headline_variants":["In curved spaces, active subspaces align to second order","Riemannian active subspaces: two views, one O(R^2) limit","Parallel transport proves active spectra match on manifolds","Active subspaces on spheres: quadratic-order agreement","Geodesic balls: intrinsic and extrinsic active subspaces concur"],"cache_read_input_tokens":37632,"weakest_assumption_plain":"The construction requires the samples to lie in a geodesic ball of radius R below the injectivity radius of the central point, so each point connects to that point by a unique geodesic and radial parallel transport is well-defined; outside such a ball the single-frame active manifold geodesics are not defined and the eigenvalue ordering loses its stated meaning.","fun_headline_variants_meta":{"raw":{"variants":["In curved spaces, active subspaces align to second order","Riemannian active subspaces: two views, one O(R^2) limit","Parallel transport proves active spectra match on manifolds","Active subspaces on spheres: quadratic-order agreement","Geodesic balls: intrinsic and extrinsic active subspaces concur"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000699,"raw_usage":{"total_tokens":2989,"prompt_tokens":737,"completion_tokens":2252,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":481,"completion_tokens_details":{"reasoning_tokens":2168}},"tokens_in":481,"tokens_out":2252,"duration_ms":16641,"temperature":1.0,"reasoning_tokens":2168,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T03:14:50.978193+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"On the unit 2-sphere, center at the north pole, sample a wrapped uniform measure on a geodesic ball of radius R, and use a function with known gradient, e.g. f(x)=a·x. Numerically evaluate ||E0^T C_ι E0 − G0||_2 for shrinking R and verify it scales as O(R^2) with the explicit constant 1+R^2/4 from Lemma 2 as R approaches the injectivity radius. A rate slower than quadratic or a violation of the stated constant would refute Proposition 3.","supporting_citations":[],"review_version":1}