{"id":"de877f69-829b-45c7-903f-9b09cf33ca6e","arxiv_id":"2505.14732","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"A surface finite element method for the p-Laplacian approximates intrinsic geodesic distances to features on surfaces, converging numerically to exact distances as p grows.","lead":"This paper uses a nonlinear PDE called the p-Laplacian to compute distances along a curved surface from a set of points or curves. The method matches true surface distances in numerical tests and handles surface boundaries more cleanly than a popular alternative.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The p→∞ convergence theorem is asserted to extend to compact Riemannian surfaces and isolated-point features, but the cited proof covers convex Euclidean domains; this unproved extension is the weakest load-bearing link in the central claim.","rationale":"The reader's weakest assumption is the unsupported extension of the p→∞ convergence theorem from convex Euclidean domains to compact Riemannian surfaces, including degenerate point features. My reading confirms this as the most load-bearing concern: the central claim of the paper is precisely that the large-p solution of the surface p-Laplacian BVP provides the intrinsic geodesic distance, and the only theoretical support cited for that limit is [10, 26], whose setting does not cover the paper's examples. The one-sentence appeal to Sobolev and Kondrachov embeddings is not a proof; those embeddings give compactness but not the comparison principle or boundary-layer control needed for uniform convergence to the distance function. The authors themselves flag that correctness is primarily numerical, which is honest but reinforces that the theoretical pillar is unproved. The numerical experiments, especially Tables 1–3 and the 1D exact solution, provide genuine supporting evidence, so I do not see grounds to reject or to move from the reader's CONDITIONAL verdict. A concrete analytical test — the radially symmetric spherical cap problem — can determine whether the point-feature setting fails in the simplest non-flat case; if it passes, the theory gap remains but is not contradicted by the available numerics. Thus the appropriate recommendation is UNCHANGED: the paper should be accepted only conditionally, pending either a rigorous proof of the surface extension or a clear statement that the claim is solely numerical, with the theoretical convergence theorem explicitly left as an open question.","tokens_in":15266,"tokens_out":9263,"duration_ms":99602,"concrete_test":"Reduce the hemisphere point-feature problem (a version of Example 4.1.1 with Γ1 a pole and Γ2 the equator) to a radial ODE. Since the solution depends only on colatitude θ, equation (2a)–(2c) becomes a two-point BVP whose solution can be written explicitly: integrate sinθ |u'_p|^{p−2}u'_p = C − (1 − cosθ) and choose C to satisfy u'_p(π/2) = 0. Take the p→∞ limit analytically and compare pointwise with the spherical distance θ. If the limit is not θ uniformly on [0, π/2], the claimed surface extension is false; if it is θ, the point-feature case survives this non-flat test, and the remaining gap is a missing general proof rather than an observed counterexample.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim — that solutions of (2a)–(2c) on a surface converge to the intrinsic geodesic distance as p→∞ — rests on [26, Remark 2.2], which is stated for smooth convex subsets of R^n. Section 2 bridges the gap with one sentence: 'the analysis in [10, 26] can be extended to compact Riemannian manifolds using the Sobolev and Kondrachov embedding theorems from [28].' Embedding theorems alone do not reproduce the comparison and barrier arguments that give uniform convergence to dist(x, Γ1) in [10, 26], especially under the natural Neumann condition on Γ2. The paper's own limitation note (Sec. 1.2: 'correctness is primarily assessed through numerical experiments') concedes this gap. The degeneracy is sharpest for Γ1 an isolated point: the authors propose replacing the point by an arbitrarily small geodesic ball, but the numerical experiments impose u = 0 at a single vertex, and no analysis is given that the p→∞ limit of the regularized problems equals the point-to-feature distance. If the correct limit on surfaces is a different ∞-harmonic function, or if point features require a different Γ2 condition, the theoretical backing for the main claim collapses; what remains is empirical evidence on a few meshes, which does currently support the claim.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes approximating intrinsic geodesic distances to feature sets on surfaces by solving a surface p-Laplacian boundary value problem with mixed Dirichlet and Neumann conditions, (2a)-(2c), for large p. The method is discretized with piecewise-linear surface finite elements and solved by ADMM. The authors present a one-dimensional exact solution, numerical convergence studies on the hemisphere and torus benchmarked against analytic geodesic distances, comparisons with the heat method and the polyhedral method, triangle-inequality checks, and robustness tests under vertex noise. The central claim is that as p approaches infinity the solution converges to the true intrinsic geodesic distance to the feature set, with numerical experiments demonstrating this convergence.","tokens_in":15557,"tokens_out":6037,"duration_ms":59592,"significance":"If the convergence claim is valid, the method provides a PDE-based alternative to heat and polyhedral geodesic distance methods, with the advantages of a natural treatment of boundary conditions on open surfaces and tunable smoothness through the parameter p. The paper's strengths include the use of external analytic benchmarks (spherical law of cosines and torus geodesic formulas), a clean one-dimensional analytic solution, a detailed ADMM formulation, and careful empirical checks of the triangle inequality and noise robustness. The reported linear convergence in 1/p on fixed meshes is convincing evidence that the numerical approach behaves as intended. However, the theoretical convergence statement for general compact Riemannian surfaces is asserted rather than proved, and the treatment of isolated-point feature sets relies on an unanalyzed regularization, so the analytical backing of the central claim is incomplete.","major_comments":[{"comment":"The paper states that the analysis in [10,26] can be extended to compact Riemannian manifolds using the Sobolev and Kondrachov embedding theorems from [28], and then asserts the limit (5). This extension is load-bearing for the central claim, but no proof is supplied, and embedding theorems alone do not reproduce the comparison and barrier arguments that yield uniform convergence to dist(x, Gamma_1) in [10,26], particularly under the natural Neumann condition on Gamma_2. Since Section 1.2 concedes that correctness is primarily assessed through numerical experiments, the authors should either provide a rigorous proof (at least for the closed-surface case and the smooth-feature case) or explicitly label the surface result as a conjecture supported by numerical evidence.","section":"Section 2, Eq. (5)"},{"comment":"For isolated-point feature sets, the paper proposes replacing the point by the boundary of an arbitrarily small geodesic ball, but the numerical experiments impose u=0 at a single vertex. No analysis is given that the p-to-infinity limit of the regularized problems equals the point-to-feature distance, and the simultaneous limits of mesh refinement and p going to infinity are not studied; Tables 1 and 2 fix a fine mesh and vary p only. To make the point-source examples rigorous, the authors should analyze the regularization or provide a numerical study of h and p tending to their limits together, explaining why the discrete point constraint does not introduce an adverse layer.","section":"Section 2, point features and Tables 1-3"},{"comment":"The justification that the natural boundary condition (4) is equivalent to the homogeneous Neumann condition (2c) is heuristic: (4) also holds when |nabla u|^{p-2}=0, and the limiting distance function itself does not satisfy (2c) on Gamma_2, as the authors themselves note in the one-dimensional example. This does not invalidate the numerical method, but the role of (2c) as a finite-p regularization and its effect on the p-to-infinity limit should be clarified, because the mismatch between the imposed finite-p boundary condition and the limiting distance function is exactly where the unproved extension to surfaces is most delicate.","section":"Section 2, Eq. (4) and (2c)"}],"minor_comments":[{"comment":"The caption contains incomplete values: 'l approx x 10^-2, h approx x 10^-2' for the hemisphere mesh, which should be filled in.","section":"Figure 7 caption"},{"comment":"The definition of S uses x>0, while Gamma_2 = partial S is later treated as part of the domain; the notation should consistently distinguish the open hemisphere from its closure.","section":"Section 4.1.2"},{"comment":"The SMAPE formula has a denominator that vanishes when both dist_i and u_i are zero (for example at the feature set); the authors should state how such vertices are handled in the reported errors.","section":"Section 4, SMAPE definition"},{"comment":"The convergence study is restricted to p-refinement on a single mesh; an h-convergence study at a fixed large p would strengthen the claim of order O(h^2 + 1/p) and clarify the interaction between geometric discretization and the p-limit.","section":"Tables 1 and 2"}],"recommendation":"major_revision","confidential_remarks":"The numerical work is solid and the paper fits the journal's scope, but the theoretical assertion of convergence on arbitrary compact Riemannian surfaces is not proven and the point-feature regularization is a genuine gap. The paper could become acceptable after a major revision in which the authors either prove the surface extension in a restricted but useful setting (e.g., smooth closed surfaces with smooth features) or recast the convergence statement as a conjecture supported by the numerical evidence, and address the point-source regularization explicitly. The manuscript's current framing risks overclaiming a theorem that is only empirically supported."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short take: this is a solid numerical methods paper. It does something new — intrinsic geodesic distance-to-feature on surfaces via the surface p-Laplacian with mixed boundary conditions — and it checks the approach properly against exact geodesics on a hemisphere and a torus, reporting clean O(1/p) convergence. The comparisons with the heat method and the polyhedral method are fair, and the triangle-inequality test is a nice addition.\n\nThe main weakness is theory. The central p→∞ convergence claim is borrowed from Euclidean convex domains and asserted, in one sentence, to extend to compact Riemannian manifolds. The authors themselves say correctness is primarily assessed through numerical experiments, so the gap is out in the open. For a numerical methods paper that is acceptable, but the one-sentence justification via Sobolev and Kondrachov embeddings is doing more work than it can support; those embeddings alone do not reproduce the comparison and barrier arguments in the Euclidean proof. Point features are similarly hand-waved: the text suggests regularizing a point by a small geodesic ball, but the experiments impose Dirichlet at a single vertex, with no analysis connecting the two.\n\nNone of this undermines the numerical evidence, which is the paper's real contribution. What is missing for reproducibility: no code or data, and the robustness-to-noise test is purely visual. The algorithmic description is detailed enough for an expert to reimplement, but shipping code would make the claim much easier to trust. The work is a good fit for a journal like Computer Aided Geometric Design. Who is it for? People working on PDE-based distance approximation on surfaces who want a method that handles Neumann boundaries naturally and offers controllable smoothness via p. It is not a speed contest; the polyhedral method is more accurate and the paper says so.\n\nI'd send this to a serious referee. The referee should ask for a reproducibility statement and, ideally, a quantitative noise experiment. The theory gap should be flagged but need not be fatal.","headline":"Solid numerical methods paper that delivers what it claims empirically, with a theory gap that is openly acknowledged; worth refereeing.","tokens_in":16045,"tokens_out":2967,"would_cite":true,"duration_ms":30260,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35J92","65N30","53C22"],"pacs":[],"model":"deepseek-v4-flash","headline":"Solving the surface p-Laplacian for large p approximates intrinsic geodesic distance to feature sets, with numerical convergence shown on hemispheres, tori, and complex meshes.","keywords":["p-Laplacian","geodesic distance","distance-to-feature","surface finite element method","ADMM","surface PDE","triangle inequality","distance estimation"],"falsifier":"Run the paper's hemisphere test at successively larger $p$ and finer meshes, comparing $u_p$ to the exact spherical distance $\\arccos(x\\cdot q)$, where $q$ is the feature point. If the $L^2$ and percentage errors stop decreasing, or if the ADMM iterates converge to a function whose level sets are not geodesic circles around the feature, the central convergence claim is refuted. A second decisive test uses a feature set that is a single isolated point on a closed surface, a case the paper flags as degenerate and does not explicitly validate.","tokens_in":15069,"feed_emoji":"📏","tokens_out":14736,"duration_ms":121505,"temperature":0.7,"pith_summary":"Geodesic distance on a surface is the shortest path measured within the surface itself, and computing it usually requires discrete graph or specialized PDE solvers. This paper claims that a single PDE, the surface $p$-Laplacian with a Dirichlet condition on the feature set and a natural Neumann condition on the remaining boundary, produces that distance: the solution $u_p$ converges to the intrinsic geodesic distance as $p\\to\\infty$. The authors carry out the computation with a surface finite element scheme and an alternating-direction optimization solver, and their numerical studies on a hemisphere and a torus show linear convergence in $1/p$ to the exact geodesic distance, along with stable contours under vertex noise and preservation of the triangle inequality. If the claim holds, it offers a variational, PDE-based route to geodesic distances that smooths with low $p$ and sharpens as $p$ grows.","feed_headline":"Large-p Laplacian recovers geodesic distance on surfaces","feed_subtitle":"A finite-element p-Poisson scheme matches exact distances and fixes heat-method boundary artifacts.","key_machinery":"The load-bearing object is the $p$-Poisson energy $$\\mathcal{E}_p(u)=\\frac{1}{p}\\int_\\$\\Omega$ |\\nabla_S u|^p\\,dA-\\int_\\$\\Omega$ u\\,dA,$$ whose Euler-Lagrange equation is the surface $p$-Laplacian boundary value problem above. The homogeneous Neumann condition on $\\Gamma_2$ arises as the natural boundary condition of this energy, which is what lets the method treat boundaries and feature sets without post-processing. The convergence mechanism is the $p\\to\\infty$ limit: minimizers of $\\mathcal{E}_p$ have gradients whose magnitude is forced toward 1, so in the limit they become the unit-gradient function with zero set $\\Gamma_1$; that function is exactly the geodesic distance. Computationally, the paper couples surface finite elements with an alternating-direction method of multipliers, introducing a slack variable $\\xi=\\nabla_S u$; each iteration alternates a one-dimensional polynomial solve for the magnitude of $\\xi$ with a linear surface Poisson solve for $u$, which keeps the severe ill-conditioning of large $p$ under control.","core_discovery":"The paper's central discovery is that the mixed boundary value problem $$-\\Delta_S^p u_p = 1 \\text{ in } \\$\\Omega$=S\\setminus\\Gamma_1,\\quad u_p=0 \\text{ on } \\Gamma_1,\\quad \\frac{\\partial u_p}{\\partial n}=0 \\text{ on } \\Gamma_2,$$ where $\\Gamma_1$ is the feature set and $\\Gamma_2$ the unconstrained boundary, has the property $\\lim_{p\\to\\infty} u_p(x)=\\mathrm{dist}(x,\\Gamma_1)$, the geodesic distance within $S$. The authors treat this as the surface analogue of the Euclidean $p$-Laplacian convergence result, asserting the Euclidean proofs extend to compact Riemannian manifolds through embedding theorems on manifolds. Their numerical evidence comes from surface finite element discretizations solved by an alternating-direction method of multipliers; errors against exact geodesic distances on a hemisphere and a torus decay linearly in $1/p$, the computed gradients approach norm one, the triangle inequality holds on their test meshes, and contours are stable under Gaussian vertex perturbations. They also position the method relative to the heat method and the polyhedral method: the $p$-Poisson distances land between the two in accuracy on their examples, with better boundary behavior than the heat method.","pith_inferences":["The paper leaves the manifold extension of the convergence theorem as an assertion; if that gap is closed, the $p$-Poisson energy would provide a variational characterization of geodesic distance as a limit of strictly convex minimizers, connecting intrinsic distance computation to optimal transport in the same way the Euclidean mixed-boundary problem connects to mass transport through a window.","A natural stress test beyond the paper: on a closed surface, take the feature set to be a single isolated point, the case the paper itself flags as degenerate, and check whether the $p\\to\\infty$ limit still equals the geodesic distance; the regularization by a small geodesic ball may need to be made explicit there.","The ADMM splitting separates the $p$-dependent scalar equation from the linear Poisson solve, which suggests a continuation strategy of solving at small $p$ and increasing $p$ in steps; the paper already uses this to initialize, but a formal cost-error analysis could make the method competitive with fast algorithms on coarse meshes.","If the triangle inequality holds beyond the tested bunny example, the $p$-Poisson distance defines a metric on the surface, making it a candidate kernel for spectral geometry or shape matching where metric axioms matter."],"forward_implications":["On open surfaces, the method computes distance-to-feature in a single solve: the natural Neumann condition replaces the ad hoc boundary handling needed by the heat method, and the paper reports improved contours near boundaries.","Since errors decay linearly in $1/p$ on fixed fine meshes, doubling the exponent roughly halves the error, and the exponent itself acts as a smoothness dial between smoothed ($p=5$) and sharp ($p=100$) distance fields.","The computed $p$-Poisson distances satisfy the triangle inequality on the tested meshes even at $p=5$, whereas the heat-method distances violate it over substantial regions; this supports using the method when a true metric is needed.","Adding Gaussian vertex noise with standard deviation up to half the average edge length changes the distance contours only minimally, so the method is usable on noisy or imperfect meshes.","Because the formulation is a surface PDE rather than a mesh graph algorithm, the paper notes it can be adapted to other surface representations such as level sets or point clouds through suitable surface PDE solvers."],"supporting_citations":[{"why":"Establishes the $p\\to\\infty$ convergence of the $p$-Laplacian boundary value problem to distance-to-boundary in Euclidean domains, the analytical template for the surface claim.","marker":"[10]"},{"why":"Proves the $p\\to\\infty$ limit for the mixed boundary value problem with Dirichlet and Neumann conditions in convex subsets of $\\mathbb{R}^n$, the direct theoretical basis for the distance-to-feature formulation.","marker":"[26]"},{"why":"Supplies the manifold embedding theorems the paper invokes to extend the Euclidean limit to compact Riemannian surfaces.","marker":"[28]"},{"why":"Provides the alternating-direction method for $p$-Laplacian distance problems and the distance-to-surface computations that this paper extends to intrinsic distance-to-feature.","marker":"[1]"},{"why":"The heat method is the primary PDE comparison; its boundary artifacts and triangle-inequality failures motivate the $p$-Poisson approach.","marker":"[2]"},{"why":"The polyhedral exact-geodesic method serves as the accuracy baseline in the numerical comparisons.","marker":"[3]"},{"why":"Gives the surface finite element error estimates for the Poisson subproblem that the alternating-direction iterations solve repeatedly.","marker":"[29]"},{"why":"Supplies the alternating-direction convergence theory used to argue the splitting iterations approach feasibility and optimality.","marker":"[31]"}],"fun_headline_variants":["p-Laplacian finite elements match geodesic distance on surfaces","Large-p p-Laplacian scheme recovers surface geodesic distances","Surface p-Laplacian distances beat heat method on boundary artifacts","Finite-element p-Laplacian: convergent, robust geodesic distances","p-Poisson large-p method: noise-tolerant, convergent geodesics"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The theoretical guarantee depends on assuming that a convergence result proved for convex subsets of Euclidean space also holds on curved surfaces with boundaries; the paper states this extension is possible but does not prove it.","fun_headline_variants_meta":{"raw":{"variants":["p-Laplacian finite elements match geodesic distance on surfaces","Large-p p-Laplacian scheme recovers surface geodesic distances","Surface p-Laplacian distances beat heat method on boundary artifacts","Finite-element p-Laplacian: convergent, robust geodesic distances","p-Poisson large-p method: noise-tolerant, convergent geodesics"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000487,"raw_usage":{"total_tokens":2397,"prompt_tokens":941,"completion_tokens":1456,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":557,"completion_tokens_details":{"reasoning_tokens":1365}},"tokens_in":557,"tokens_out":1456,"duration_ms":12425,"temperature":1.0,"reasoning_tokens":1365,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T20:09:09.034175+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the paper's hemisphere test at successively larger $p$ and finer meshes, comparing $u_p$ to the exact spherical distance $\\arccos(x\\cdot q)$, where $q$ is the feature point. If the $L^2$ and percentage errors stop decreasing, or if the ADMM iterates converge to a function whose level sets are not geodesic circles around the feature, the central convergence claim is refuted. A second decisive test uses a feature set that is a single isolated point on a closed surface, a case the paper flags as degenerate and does not explicitly validate.","supporting_citations":[{"cited_title":"Bhattacharya, E","cited_arxiv_id":null,"evidence_quote":"Establishes the $p\\to\\infty$ convergence of the $p$-Laplacian boundary value problem to distance-to-boundary in Euclidean domains, the analytical template for the surface claim."},{"cited_title":"Garcia-Azorero, J","cited_arxiv_id":null,"evidence_quote":"Proves the $p\\to\\infty$ limit for the mixed boundary value problem with Dirichlet and Neumann conditions in convex subsets of $\\mathbb{R}^n$, the direct theoretical basis for the distance-to-feature formulation."},{"cited_title":"Aubin, Nonlinear Analysis on Manifolds","cited_arxiv_id":null,"evidence_quote":"Supplies the manifold embedding theorems the paper invokes to extend the Euclidean limit to compact Riemannian surfaces."},{"cited_title":"Fayolle, A","cited_arxiv_id":null,"evidence_quote":"Provides the alternating-direction method for $p$-Laplacian distance problems and the distance-to-surface computations that this paper extends to intrinsic distance-to-feature."},{"cited_title":"Crane, C","cited_arxiv_id":null,"evidence_quote":"The heat method is the primary PDE comparison; its boundary artifacts and triangle-inequality failures motivate the $p$-Poisson approach."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The polyhedral exact-geodesic method serves as the accuracy baseline in the numerical comparisons."},{"cited_title":"Dziuk, C","cited_arxiv_id":null,"evidence_quote":"Gives the surface finite element error estimates for the Poisson subproblem that the alternating-direction iterations solve repeatedly."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the alternating-direction convergence theory used to argue the splitting iterations approach feasibility and optimality."}],"review_version":1}