{"id":"2a3c591a-e353-4c75-b7d9-6b776c1a33c9","arxiv_id":"2606.07440","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":3.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Defines differentiable weak distance on SE(3) for surface measures via Sobolev norms and shows local optimization with trust-region methods and NUFFT gradients.","lead":"The paper defines a weak distance between compact surfaces using negative-order Sobolev norms on their measures and optimizes it over rigid motions on the special Euclidean group via gradient-based methods with NUFFT acceleration. A smart generalist might read it for insights into efficient shape registration or alignment in 3D geometry applications.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.3","headline":"Differentiability of squared Sobolev distance on SE(3) requires Fourier decay of surface measures sufficient to absorb |xi| factors from group derivatives; general compact surfaces may lack this without extra regularity assumptions.","rationale":"The reader's weakest assumption directly identifies the differentiability step as load-bearing; the concern above is a precise technical risk to that step. Full-text verification of surface regularity and the precise range of s would settle it. No other internal inconsistency is visible from the given material.","tokens_in":1687,"tokens_out":441,"duration_ms":19598,"concrete_test":"Extract the explicit gradient formula (likely derived after the Plancherel representation) and verify absolute integrability of the differentiated integrand over ξ for the minimal s that makes the original norm finite; if the integral diverges on a positive-measure set of frequencies for any compact C^0 surface, the differentiability claim does not hold.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim requires that f(g)^2 with f(g) = ||g·μ - ν||_{H^{-s}}^2 is C^1 (or higher) on SE(3) for suitable s. Via Plancherel this is an integral ∫ |e^{-i ξ·t} hat{μ}(R^T ξ) - hat{ν}(ξ)|^2 (1+|ξ|^2)^{-s} dξ. Differentiating w.r.t. rotation parameters acts on both the phase and the argument R^T ξ, producing extra |ξ| multipliers in the integrand. If the surfaces are merely compact and embedded (no C^2 or higher regularity stated), hat{μ} decays at most like |ξ|^{-1} (or slower), so the differentiated integrand may fail to be integrable precisely when s is chosen only large enough for f itself to be finite. This would invalidate the derivative-based framework and the NUFFT gradient structure.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript defines a weak distance between two compact surfaces in R^3 by identifying them with their surface measures and measuring discrepancy via the inhomogeneous Sobolev norm of negative order, realized through the Plancherel theorem. The distance is lifted to a function on the special Euclidean group SE(3) by letting group elements act via pushforward of measures. The central claim is that, for suitable negative exponents s, the square of this function is sufficiently differentiable on SE(3) to permit derivative-based local optimization; moreover, the resulting gradient admits an efficient NUFFT realization. Numerical experiments apply the SR1 trust-region method to several root-finding problems on SE(3) and note a connection to a geometric quantity.","tokens_in":1894,"tokens_out":724,"duration_ms":15549,"significance":"If the differentiability claim holds with the stated regularity, the work supplies a mathematically coherent and computationally attractive route to rigid registration that exploits the Fourier structure of weak Sobolev distances. The NUFFT gradient structure is a concrete practical advantage, and the numerical demonstration of convergence on root-finding tasks provides initial evidence of utility. The approach sits at the intersection of harmonic analysis, Lie-group optimization, and geometric measure theory; successful verification would make it relevant to both theoretical and applied communities working on surface alignment.","major_comments":[{"comment":"Abstract and §2 (definition of the objective): the claim that f(g)^2 with f(g) = ||g·μ − ν||_{H^{-s}}^2 is C^1 (or higher) on SE(3) for 'appropriate' s is asserted without a derivation showing that the differentiated integrand remains integrable. Differentiating with respect to rotation parameters produces additional |ξ| multipliers; for merely compact embedded surfaces, |hat μ(ξ)| decays at best like |ξ|^{-1}, so integrability after differentiation is not automatic when s is chosen only large enough to make the original norm finite. This directly affects the validity of the derivative-based framework.","section":"Abstract and §2"},{"comment":"§3 (gradient derivation): the favorable NUFFT structure of the gradient is presented, but the passage from the Plancherel representation to the explicit gradient formula does not include the dominated-convergence or integrability argument needed to justify differentiation under the integral sign when the surfaces have only the regularity implicit in being compact and embedded.","section":"§3"},{"comment":"Numerical section: the reported convergence of the SR1 trust-region method is observed on 'a few root-finding problems,' yet no table or figure quantifies the achieved residual, the chosen exponent s, the surface regularity, or a comparison against a non-differentiable baseline, leaving the practical payoff of the differentiability claim unverified.","section":"Numerical experiments"}],"minor_comments":[{"comment":"Notation for the inhomogeneous Sobolev norm H^{-s} should be introduced with an explicit formula (e.g., the precise weight (1+|ξ|^2)^{-s}) at first use rather than only via Plancherel.","section":"§2"},{"comment":"The connection between the optimized distance and the 'more geometric quantity' mentioned in the abstract is stated but not quantified; a short paragraph or plot relating the two would clarify the geometric interpretation.","section":"Numerical experiments"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the careful and constructive report. The comments correctly identify places where additional rigor and quantitative detail are needed. We respond to each major comment below and will revise the manuscript accordingly.","responses":[{"response":"We agree that the manuscript asserts C^1 regularity for appropriate s without supplying the explicit integrability argument. In the revision we will add a lemma in §2 that verifies differentiation under the integral is justified for s > 2. The proof will use the known |ξ|^{-1} decay of the Fourier transform of compact embedded surface measures together with the |ξ|^{-2s} weight to produce a dominating integrable function after the extra |ξ| factor from differentiating the rotation action appears.","revision_made":"yes","referee_comment":"[Abstract and §2] Abstract and §2 (definition of the objective): the claim that f(g)^2 with f(g) = ||g·μ − ν||_{H^{-s}}^2 is C^1 (or higher) on SE(3) for 'appropriate' s is asserted without a derivation showing that the differentiated integrand remains integrable. Differentiating with respect to rotation parameters produces additional |ξ| multipliers; for merely compact embedded surfaces, |hat μ(ξ)| decays at best like |ξ|^{-1}, so integrability after differentiation is not automatic when s is chosen only large enough to make the original norm finite. This directly affects the validity of the derivative-based framework."},{"response":"We concur that an explicit dominated-convergence justification is absent. The revised §3 will include a short paragraph (or appendix reference) that invokes the same integrability estimates established in the new §2 lemma to justify passing the derivative inside the Plancherel integral, thereby rigorously supporting the NUFFT gradient formula.","revision_made":"yes","referee_comment":"[§3] §3 (gradient derivation): the favorable NUFFT structure of the gradient is presented, but the passage from the Plancherel representation to the explicit gradient formula does not include the dominated-convergence or integrability argument needed to justify differentiation under the integral sign when the surfaces have only the regularity implicit in being compact and embedded."},{"response":"The numerical section will be expanded. We will add a table listing, for each example, the final residual norm, the value of s employed, a statement of the assumed surface regularity, and a short comparison of iteration counts against a derivative-free optimizer on the same instances. This will make the practical benefit of the differentiable formulation explicit.","revision_made":"yes","referee_comment":"[Numerical experiments] Numerical section: the reported convergence of the SR1 trust-region method is observed on 'a few root-finding problems,' yet no table or figure quantifies the achieved residual, the chosen exponent s, the surface regularity, or a comparison against a non-differentiable baseline, leaving the practical payoff of the differentiability claim unverified."}],"tokens_in":1508,"tokens_out":640,"duration_ms":24017,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The core contribution is a concrete way to turn surface alignment into an optimization problem over SE(3) by treating the surfaces as measures and measuring their difference in an inhomogeneous Sobolev norm of negative order. The squared distance is treated as a function on the group, its gradient is derived in Fourier space, and NUFFT is proposed to evaluate it efficiently. Numerical runs with an SR1 trust-region solver on a handful of test cases show convergence to local minima, and the authors note a link to a geometric quantity.\n\nThis is a targeted extension that puts existing Fourier and Lie-group tools into one place for this specific task. The setup is clean on paper and the NUFFT observation is practical.\n\nThe soft spot is the differentiability claim. The abstract says that for suitable exponents the square becomes differentiable enough for derivative-based search. Differentiating with respect to rotations brings extra |ξ| factors into the integrand; if the surfaces are only compact and embedded, their Fourier transforms decay slowly enough that integrability can fail exactly when the norm itself is finite. The paper would need to state the precise regularity on the surfaces or prove the decay is sufficient. Without that, the derivative framework rests on an assumption that may not hold for the stated class of objects.\n\nThe experiments are too thin to test robustness. This work is mainly for researchers already doing manifold optimization or shape matching with weak metrics. It is coherent enough to go to a referee who can check the Fourier estimates and the numerical implementation, but it is not yet ready for a broad audience.","headline":"Paper sets up a Sobolev weak distance between surface measures on SE(3) and optimizes its square locally via gradients and NUFFT, with basic numerical checks; differentiability holds only under extra regularity not fully spelled out for arbitrary compact surfaces.","tokens_in":2372,"tokens_out":408,"would_cite":false,"duration_ms":20143,"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":"The square of a Sobolev discrepancy between surface measures becomes differentiable on the special Euclidean group for appropriate exponents, enabling gradient-based local optimization with NUFFT-efficient gradients.","keywords":["surface measures","Sobolev norm","special Euclidean group","local optimization","nonuniform fast Fourier transform","trust-region method"],"falsifier":"A concrete pair of surfaces and exponents where the squared distance fails to be twice differentiable, or where the gradient cannot be written in a form that permits nonuniform fast Fourier transform evaluation, would falsify the central claim.","tokens_in":2568,"feed_emoji":"","tokens_out":649,"duration_ms":14671,"temperature":0.7,"pith_summary":"The paper turns alignment of two compact surfaces under rigid motions into an optimization problem on the special Euclidean group. Surfaces are represented by their associated measures and mismatch is measured by an inhomogeneous Sobolev norm of negative order whose square, via the Plancherel theorem, becomes a function on the group. For suitable exponents this squared function gains enough differentiability for derivative-based local search. Its gradient takes a form that supports fast numerical evaluation by the nonuniform fast Fourier transform. Experiments show the SR1 trust-region method converges on several root-finding instances and link the analytic quantity to a geometric registration measure.","feed_headline":"Squared Sobolev distance on SE(3) becomes differentiable for optimization","feed_subtitle":"Appropriate negative-order exponents make the squared mismatch differentiable enough for gradient-based search on rigid motions, with gradie","key_machinery":"the squared inhomogeneous Sobolev distance of negative order between a surface measure and its pushforward under group elements, viewed as a function on the special Euclidean group","core_discovery":"Identifying compact surfaces with their surface measures and quantifying discrepancy by the inhomogeneous Sobolev norm of negative order via the Plancherel theorem produces a distance that can be viewed as a function on the special Euclidean group. For appropriate exponents the square of this function is sufficiently differentiable to support derivative-based local minimization, and the gradient admits an efficient implementation through the nonuniform fast Fourier transform.","pith_inferences":["The same construction could be tested on other compact Lie groups that act by pushforward on measures.","Varying the Sobolev exponent may trade the guaranteed differentiability against sensitivity to high-frequency surface features.","Convergence observed in the experiments suggests the objective landscape may possess additional structure not required by the differentiability proof."],"forward_implications":["Local minima of surface discrepancy under rigid motions can be located by derivative-based solvers such as the SR1 trust-region method.","Gradient evaluations of the objective become practical through existing nonuniform fast Fourier transform routines.","The analytic distance connects directly to a geometric registration quantity in the reported numerical tests."],"fun_headline_variants":["Squared Sobolev distance differentiable for SE(3) optimization","Negative Sobolev distance squared for local optimization on SE(3)","Surface discrepancy on SE(3) made differentiable via Sobolev norm","Local optimization of weak distance possible on special Euclidean group"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"Identifying surfaces with surface measures and quantifying discrepancy via the inhomogeneous Sobolev norm of negative order yields a distance whose square is sufficiently differentiable on the Lie group for the chosen exponents, with the isometry action corresponding exactly to pushforward of measures.","fun_headline_variants_meta":{"raw":{"variants":["Squared Sobolev distance differentiable for SE(3) optimization","Negative Sobolev distance squared for local optimization on SE(3)","Surface discrepancy on SE(3) made differentiable via Sobolev norm","Local optimization of weak distance possible on special Euclidean group"]},"model":"grok-4.3","cost_usd":0.005974,"raw_usage":{"total_tokens":2714,"prompt_tokens":595,"num_sources_used":0,"completion_tokens":67,"cost_in_usd_ticks":59740500,"prompt_tokens_details":{"text_tokens":595,"audio_tokens":0,"image_tokens":0,"cached_tokens":64},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2052,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":595,"tokens_out":67,"duration_ms":10623,"temperature":1.0,"reasoning_tokens":2052,"cache_read_input_tokens":64,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-27T21:04:58.247210+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A concrete pair of surfaces and exponents where the squared distance fails to be twice differentiable, or where the gradient cannot be written in a form that permits nonuniform fast Fourier transform evaluation, would falsify the central claim.","supporting_citations":[],"review_version":1}