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REVIEW 3 major objections 4 minor 37 references

Geodesic distance approximation using a surface finite element method for the $p$-Laplacian

T0 review · 3 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict Solid numerical methods paper that delivers what it claims empirically, with a theory gap that is openly acknowledged; worth refereeing. read the letter →

arxiv 2505.14732 v1 pith:LAAMJDSJ submitted 2025-05-20 cs.GR cs.NAmath.NA

Hannah Potgieter , Razvan C. Fetecau , Steven J. Ruuth This is my paper · ORCID
classification cs.GRcs.NAmath.NA MSC 35J9265N3053C22
keywords p-Laplaciangeodesicdistancedistance-to-featuresurfacefiniteelementmethodADMMPDEtriangleinequalityestimation
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

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.

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 (3)
  1. [Section 2, Eq. (5)] 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.
  2. [Section 2, point features and Tables 1-3] 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.
  3. [Section 2, Eq. (4) and (2c)] 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.
minor comments (4)
  1. [Figure 7 caption] The caption contains incomplete values: 'l approx x 10^-2, h approx x 10^-2' for the hemisphere mesh, which should be filled in.
  2. [Section 4.1.2] 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.
  3. [Section 4, SMAPE definition] 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.
  4. [Tables 1 and 2] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the p-Laplacian limit result and numerical benchmarks are external to the method, and the paper's unproved surface extension is a support gap, not a circular reduction.

full rationale

The paper's central claim is that solutions of the surface p-Laplacian BVP (2a)-(2c) approximate intrinsic geodesic distance as p grows. This claim is not derived from the target distance by construction: the PDE is a known variational problem, and the p→∞ convergence to dist(x, Γ1) is imported from external results by Bhattacharya et al. [10] and Garcia-Azorero et al. [26], which are neither authored by the present authors nor defined in terms of the geodesic distance they seek. The numerical validation compares u_p against independent analytical geodesic formulas (spherical law of cosines for the hemisphere, explicit torus distances), so the reported accuracy is not fitted into existence. The parameter p is a mathematical limit parameter rather than a fitted constant, and no term in the ADMM objective encodes the benchmark distances. The only notable weakness is the paper's assertion that the Euclidean-domain analysis of [10, 26] extends to compact Riemannian manifolds via Sobolev and Kondrachov embeddings from [28]; this is stated without proof and the paper itself concedes that 'correctness is primarily assessed through numerical experiments.' That is an unverified assumption and a correctness risk, but it is not circularity, because the cited results do not presuppose the surface geodesic conclusion. The self-citation to [30] is incidental and not load-bearing. No step in the derivation reduces by construction to its own input, so a score of 0 is appropriate.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central claim rests on one proved external theorem (Euclidean mixed-BC limit), one asserted manifold extension, a heuristic for the Neumann boundary condition, a regularization for degenerate features, and an assumed ADMM convergence on surfaces. None of these are derived in the paper; the only independent support is the 1D exact solution and the numerical convergence studies.

free parameters (3)
  • p (p-Laplacian exponent) = varied 5 to 160 in tests; no automatic selection
    User-chosen parameter controlling the approximation; large p approaches the exact geodesic distance. Not fitted to the target distances but influences accuracy.
  • β (ADMM penalty parameter) = 10 (typical)
    Chosen to balance primal and dual residuals; the paper notes smaller β gives a better dual residual and larger β gives a better primal residual.
  • ADMM stopping tolerances = tol_primal=1e-6, tol_dual=1e-3
    Algorithmic stopping criteria, not central to the claim but needed to reproduce the reported iteration counts.
assumptions (5)
  • standard math For a bounded domain Ω in R^n with mixed boundary conditions, the solution u_p of the p-Laplacian BVP converges uniformly to dist(x, Γ1) as p→∞.
    Proved in [26] (cited); the paper builds on this result.
  • domain assumption This limit result extends to compact Riemannian surfaces, e.g., via Sobolev and Kondrachov embedding theorems from [28].
    Asserted in Section 2 without proof; load-bearing for the surface setting of the central claim.
  • ad hoc to paper On Γ2, the natural boundary condition is equivalent to the homogeneous Neumann condition for finite p because |∇u_p| is close to 1 for large p.
    Heuristic justification in Section 2; the 1D exact solution supports it, but it is not a rigorous general argument.
  • domain assumption A feature set containing isolated points can be treated as the boundary of an arbitrarily small geodesic ball; the solution converges to the point-to-distance function as the ball shrinks.
    Stated in Section 2 to handle degenerate Γ1; not proved.
  • domain assumption ADMM converges for the surface p-Poisson BVP; the convergence analysis from [32, 33] (plane) carries over to surfaces.
    The paper says 'Provided these assumptions hold' and 'Our numerical simulations exhibit convergence' (Section 3); the surface case is not analyzed.

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Pith. "Pith review of Geodesic distance approximation using a surface finite element method for the $p$-Laplacian." pith.science (2026). https://pith.science/paper/LAAMJDSJ

@misc{pith2026250514732,
  author       = {Pith},
  title        = {Pith review of: Geodesic distance approximation using a surface finite element method for the $p$-Laplacian},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LAAMJDSJ}},
  note         = {Machine review of arXiv:2505.14732}
}
abstract

We use the $p$-Laplacian with large $p$-values in order to approximate geodesic distances to features on surfaces. This differs from Fayolle and Belyaev's (2018) [1] computational results using the $p$-Laplacian for the distance-to-surface problem. Our approach appears to offer some distinct advantages over other popular PDE-based distance function approximation methods. We employ a surface finite element scheme and demonstrate numerical convergence to the true geodesic distance functions. We check that our numerical results adhere to the triangle inequality and examine robustness against geometric noise such as vertex perturbations. We also present comparisons of our method with the heat method from Crane et al. [2] and the classical polyhedral method from Mitchell et al. [3].

Figures

Figures reproduced from arXiv: 2505.14732 by the authors.

Figure 1
Figure 1. S is the open hemisphere and Γ1 (indicated in orange) represents the feature set. Top row: Distance-to-boundary problem (Γ1 = ∂S, Γ2 = ∅). Bottom row: Distance-to-feature problem. Γ2 is indicated in blue. general case when Γ1 is a strict subset of ∂Ω (hence, Γ2 ̸= ∅). Specifically, the authors of [26] considered the mixed BVP (2a)-(2c) on smooth convex subsets of R n , and investigated the limit p → ∞ of its weak so… view at source ↗
Figure 2
Figure 2. One-dimensional example, with S = (−1, 1) and Γ1 = {0}. Left: the exact solution up(x) (see (6a)-(6c) and (7)) converges to |x| = dist(x, Γ1) as p increases. Right: the corresponding pointwise errors |x| − up(x), for various p-values. where P 1 is the space of linear elements on Th, the collection of faces on the surface mesh. The corresponding lifted finite element space is S l h = {φh = ϕ l h : ϕh ∈ Sh}, where the… view at source ↗
Figure 3
Figure 3. Computed distance approximations using three methods: our [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: Tracking of residuals and gradient norm for the example of distance to a point on the hemisphere [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]
Figure 5
Figure 5. Figure 5: p-Poisson energy (3) versus ADMM iterations for distance to closed curves on torus using various p-values. The algorithm is run on a torus mesh with 196608 tri cells and ℓ ≈ 3.30 × 10−2 , h ≈ 5.49 × 10−2 . SMAPE produced by the heat method is comparable to that of the …
Figure 6
Figure 6. Figure 6: Computed distance approximations using three methods: the [PITH_FULL_IMAGE:figures/full_fig_p016_6.png]
Figure 7
Figure 7. Figure 7: Normalized L 1 norm of 1 − [PITH_FULL_IMAGE:figures/full_fig_p018_7.png]
Figure 8
Figure 8. Figure 8: Triangle inequality check. The feature set consists of a point [PITH_FULL_IMAGE:figures/full_fig_p019_8.png]
Figure 9
Figure 9. Figure 9: Computed p-Poisson distances to the lid of a teapot with Gaussian noise added to the mesh vertices. The standard deviation is in terms of the average mesh cell edge length ℓ ≈ 1.60 × 10−2 . Top left: no noise. Top right: minimal amount of noise (σ = ℓ 8 ). Bottom left:…

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