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REVIEW 3 major objections 6 minor 74 references

Extracting Complex Topology from Multivariate Functional Approximation: Contours, Jacobi Sets, and Ridge-Valley Graphs

T0 review · 3 major / 6 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read This paper presents the first framework for extracting contours, Jacobi sets, and ridge-valley graphs directly from multivariate functional approximation models, without discretizing the domain.

desk verdict First direct extraction of contours, Jacobi sets, and ridge-valley graphs from MFA models, with plausible but partly unvalidated heuristics; worth refereeing but needs stronger correctness evidence and released code. read the letter →

arxiv 2508.07637 v1 pith:RRAIVYLR submitted 2025-08-11 cs.LG cs.CG

Guanqun Ma , David Lenz , Hanqi Guo , Tom Peterka , Bei Wang This is my paper · ORCID
classification cs.LGcs.CG
keywords multivariatefunctionalapproximationcontinuousimplicitmodelstopologicaldataanalysiscontourextractionJacobisetsridge-valleygraphsparticletracingB-splinesurfaces
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

Modern scientific data is increasingly stored in continuous surrogates rather than grids. This paper tries to show that such a surrogate, specifically a multivariate functional approximation (MFA) built from piecewise-polynomial B-splines, can be analyzed topologically on its own terms: contours, Jacobi sets, and ridge-valley graphs are all extractable directly from the model's value and derivative queries. The method traces a contour by moving along the direction perpendicular to the gradient, and reduces Jacobi-set and ridge-valley extraction to contour extraction of derived functions. If the approach holds, continuous implicit models become first-class inputs for topological data analysis and visualization, avoiding the zigzag artifacts that appear when the model is first discretized.

What carries the argument

The load-bearing device is the reduction of every descriptor to a contour problem, plus particle tracing along tangent directions. A contour of $f$ at value $v$ follows the unit tangent $t = (-f_y, f_x) / \lVert \nabla f \rVert$; the Jacobi set of $(f,g)$ is the zero contour of the comparison function $c = f_x g_y - f_y g_x$; a ridge-valley graph is the zero contour of $c$ for $(f, \lVert \nabla f \rVert^2)$. Since MFA spans are polynomial, all derived functions remain polynomial on the same spans, so one RK4 contour tracer with normalized-gradient-descent seeding and a projection correction step handles all three extractions; critical points are appended from Newton-based extraction on the

What would settle it

Build a smooth function whose level set contains a small closed loop lying entirely inside one MFA span, place the uniform seed grid so that no seed lies in the loop's basin of attraction under normalized gradient descent, and check whether the reported #Loop and #CC silently drop that loop.

Watch

Extended reading notes

Core claim

On its own terms, the paper's discovery is that three standard topological descriptors do not require a discrete scaffold when the input function is continuous and differentiable. For a single function $f$, the level set $\{f = v\}$ is traced as an integral curve of the tangent field perpendicular to $\nabla f$, with normalized gradient descent supplying initial points inside each polynomial span and RK4 integration carrying the curve; critical points are inserted separately. For a pair $(f,g)$, the Jacobi set is identified with the zero level set of the cross product $f_x g_y - f_y g_x$, and a ridge-valley graph is treated as the Jacobi set of $f$ and $\lVert \nabla f \rVert^2$; both become

Load-bearing premise

The argument assumes that seeding a fixed number of starting points per polynomial span and running normalized gradient descent lands on every separate piece of a contour in that span, so no contour segment is silently missed.

Editorial extensions

If this is right

  • One contour-extraction implementation, with derived functions swapped in, yields all three descriptors, unifying the main building blocks of scalar and multifield topology.
  • Topological fidelity no longer depends on grid density: users can refine features by changing step size and tolerance rather than resampling the whole domain.
  • On synthetic contour experiments the method matches ground-truth loop and connected-component counts with pointwise errors near the chosen accuracy threshold.
  • On scientific Jacobi-set and ridge-valley experiments, the continuous extraction reports far fewer spurious loops than the discretization-based baselines, which exhibit zigzag artifacts.
  • Because only value, gradient, and Hessian queries are needed, the method is portable to other continuous implicit models with the same query interface.

Reading between the lines

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

  • The authors do not test this, but when the method is applied to neural implicit models, derivative queries become approximate rather than exact; a natural experiment is to track how loop and component counts degrade as derivative error grows.
  • The same zero-contour reduction may extend to other restricted-criticality descriptors such as Pareto sets or Reeb-space singularities, since those are also characterized by aligned gradients; the paper does not claim this extension.
  • A cheap audit follows from the paper's own seeding assumption: rerun extraction with randomly shifted seed grids inside each span and measure the variance of the reported #Loop and #CC; large variance would expose silently missed contour pieces.
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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 / 6 minor

Summary. The paper presents a framework for extracting contours, Jacobi sets, and ridge-valley graphs directly from multivariate functional approximation (MFA) B-spline models, without first discretizing the domain. Contour extraction is cast as particle tracing along the tangent direction of the level set: normalized gradient descent finds starting points, RK4 integration traces trajectories, and a correction step keeps points on the level set; trajectories are merged and connected near critical points. Jacobi sets and ridge-valley graphs are reduced to zero-contour extraction of derived functions h and rho, respectively. Experiments on synthetic and scientific MFA models compare the continuous method with discrete pipelines (ParaView/TTK), reporting max/avg residuals and loop/component counts.

Significance. If the method is sound, it is a useful step toward topological analysis of implicit continuous models, avoiding aliasing artifacts introduced by full-domain discretization. The reduction of Jacobi and ridge-valley extraction to contour extraction is conceptually elegant and well connected to existing definitions (Edelsbrunner-Harer, Norgard-Bremer). The synthetic contour experiments (Sinc, Schwefel) match ground-truth loop and component counts, and the paper demonstrates parallel efficiency. However, the completeness of the initialization step is an unproven assumption that is load-bearing for the main claims, and the reported error metrics are partially tautological because the correction step enforces the level-set condition. The experimental validation for Jacobi sets and ridge-valley graphs also lacks analytic ground truth, leaving false-positive/false-negative rates undetermined.

major comments (3)
  1. [Sec. 4.1.1] The completeness of contour extraction rests on the assumption that (p+1)^d uniformly placed seeds per span, followed by normalized gradient descent (Eq. 13), find at least one starting point on every contour piece. This is explicitly stated as an assumption in Supplement B, and the Limitations section concedes that initial point selection may affect results. The number of seeds depends only on polynomial degree, not on the number or geometry of contour components in a span; a degree-4 span can contain multiple disjoint closed contours, and nothing guarantees that each component's basin contains a seed. A missed component silently changes #Loop and #CC, which are the headline metrics, and the failure propagates to Jacobi/ridge extractions because Secs. 4.2 and 4.3 reduce them to contours of higher-degree derived functions. No coverage analysis or empirical study of missed components is p
  2. [Sec. 4.1.2] The reported max/avg errors are partly by construction: every traced point is projected back onto the level set whenever |f(x)-c| > epsilon using the same normalized gradient descent (Eq. 13). Thus e_max and e_avg below epsilon are residuals of a constraint-satisfaction step, not independent measures of geometric accuracy. The same applies to h for Jacobi sets and rho for ridge-valley graphs. This does not invalidate the method, but the error tables should be labeled as residuals, and the comparison with the discrete method should be supplemented by an independent accuracy measure (e.g., distance to an analytic Jacobi set for the Gaussian-pair model).
  3. [Sec. 5.4] For Jacobi sets and ridge-valley graphs, no analytic ground truth is used; the continuous method is only compared with the discrete PL method, which is known to produce zigzag artifacts and spurious loops. The large #Loop counts of the discrete method are therefore not evidence that the continuous method's #Loop/#CC are correct. A synthetic model with known Jacobi set and a ridge-valley example with known graph should be added to assess false positives and false negatives. Without this, the central claim of 'accuracy' for these descriptors is underdetermined.
minor comments (6)
  1. [Sec. 3.2] The Newton update formula appears garbled in the typeset text; it should show the inverse Hessian applied to the gradient. Please ensure the equation is rendered correctly.
  2. [Sec. 4.4] The time-complexity expression is difficult to parse due to symbol corruption. Please typeset it cleanly and define each symbol (n, L, T_g, etc.) in the surrounding text.
  3. [Sec. 4.1.2] The symbol h is used both for the RK4 step size (Eq. 12) and for the Jacobi-set derived function (Eq. 15). This is confusing; suggest using a different symbol for one of them.
  4. [Supplement B] The degree formulas for h and rho are garbled in the text. They should be stated explicitly, e.g., deg(h)=deg(f)+deg(g)-2 and the corresponding formula for rho, so that the number of initial points is unambiguous.
  5. [Sec. 5] The column headers 'Step Size...' and 'Sampling Ratio' are not clearly separated; the sampling ratio applies only to the discrete method. Please clarify in the caption.
  6. [Sec. 5.2] The statement that 'the errors at all nodes are below the threshold' should make clear that this is guaranteed by the correction step; without this clarification, a reader may interpret it as an independent validation.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: the reductions to contour extraction are exact external identities; the seeding assumption is a completeness risk, not circularity.

full rationale

The paper's derivation chain is self-contained. Contour extraction is a numerical procedure: points on f=c are found by normalized gradient descent (Eq. 13) and traced by RK4 along the tangent direction (Eqs. 11–12); no output quantity is used to define an input. Jacobi set extraction uses the known mathematical equivalence J(f,g) = zero level set of det[grad f, grad g] (Eqs. 8, 14, 15), and ridge-valley graph extraction uses the external definition from Norgard and Bremer [43] as the Jacobi set of f and ||grad f||^2 (Eqs. 17–21). These are exact identities, not renamed predictions or fitted inputs. The only self-citation is [34] for critical-point extraction (Sec. 3.2), used as a subroutine to insert critical points into contours; [34] is peer-reviewed, independently published, and not the target result, so it is real evidence rather than load-bearing circularity. The paper itself flags the main weakness in Sec. 6 Limitations: 'the selection of initial points ... may affect the results ... left for future work,' and Supplement B states 'We assume that this sampling strategy allows us to find at least one starting point on each trajectory through gradient descent.' This is an unproven completeness assumption, which is a correctness risk (missed contour components would silently change #Loop/#CC and propagate to Jacobi and ridge-valley extraction), but it is not circular: it is an input assumption, not a conclusion derived from itself. Likewise, choosing the step size until #Loop/#CC converge and then reporting #Loop/#CC is a mild self-referential evaluation practice, but the synthetic models (Sinc, Schwefel) are checked against closed-form ground truth (Tabs. 3, 9), so the central claims do not reduce to a fit.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central claim rests on standard B-spline theory and on the Morse/genericity of the represented functions. It also depends on two ad hoc heuristics: the initial-point sampling assumed to seed every contour piece, and the endpoint-connection threshold assumed to preserve topology. Three parameters (step size, accuracy threshold, connection threshold) are tuned per model via convergence studies; the accuracy threshold additionally makes the reported residual errors partially self-confirming.

free parameters (3)
  • Particle tracing step size h = 1e-2 for Schwefel, Sinc, Gaussian Pair, Gaussian Mixture; 1e-4 for S3D, Karman, Boussinesq, CESM; 1e-5 for Hurricane
    Selected per model-task pair via ablation convergence of #Loop and #CC on the same datasets later used for evaluation (Supplement B, Tab. 8).
  • Accuracy threshold epsilon = 1e-9 for all experiments
    Chosen by evaluating values from 1e-7 to 1e-10 across models. The correction step enforces |f-c| < epsilon, making reported errors below this value a consequence of the algorithm rather than an independent measurement.
  • Trajectory connection threshold tau = 1e-2 for all experiments
    Chosen via evaluation of values from 1e-4 to 1e-1 on synthetic models to achieve correct connectivity (Supplement B).
assumptions (4)
  • standard math B-spline basis functions provide exact function values and derivatives of any order at any point within a span.
    Relied on throughout Secs. 4.1-4.3 for exact gradient and Hessian queries; standard B-spline theory from de Boor and Piegl-Tiller.
  • domain assumption The functions f and h are Morse functions, or generic enough that Jacobi sets are embedded 1-manifolds and contour critical points have valence up to 4.
    Sec. 4.1 opens with 'Given a Morse function f'; Sec. 3.4's Jacobi set definition assumes Morse functions. Real MFA fits may not strictly satisfy Morse conditions, affecting the required valence and manifold structure.
  • ad hoc to paper Uniformly sampling (p+1)^d initial points per span and applying normalized gradient descent finds at least one starting point on every contour piece in the span.
    Supplement B, 'Number of initial points': 'We assume that this sampling strategy allows us to find at least one starting point on each trajectory through gradient descent.' The Limitations section states no guarantee is provided.
  • ad hoc to paper Connecting trajectory endpoints within distance tau preserves the correct topology, creating neither false loops nor spurious merges.
    Sec. 4.1.3 connection steps rely on the threshold tau; the Limitations section admits 'our connecting strategy may introduce incorrect connections when trajectories are close enough.'

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Cite this review

Pith. "Pith review of Extracting Complex Topology from Multivariate Functional Approximation: Contours, Jacobi Sets, and Ridge-Valley Graphs." pith.science (2026). https://pith.science/paper/RRAIVYLR

@misc{pith2026250807637,
  author       = {Pith},
  title        = {Pith review of: Extracting Complex Topology from Multivariate Functional Approximation: Contours, Jacobi Sets, and Ridge-Valley Graphs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RRAIVYLR}},
  note         = {Machine review of arXiv:2508.07637}
}
read the original abstract

Implicit continuous models, such as functional models and implicit neural networks, are an increasingly popular method for replacing discrete data representations with continuous, high-order, and differentiable surrogates. These models offer new perspectives on the storage, transfer, and analysis of scientific data. In this paper, we introduce the first framework to directly extract complex topological features -- contours, Jacobi sets, and ridge-valley graphs -- from a type of continuous implicit model known as multivariate functional approximation (MFA). MFA replaces discrete data with continuous piecewise smooth functions. Given an MFA model as the input, our approach enables direct extraction of complex topological features from the model, without reverting to a discrete representation of the model. Our work is easily generalizable to any continuous implicit model that supports the queries of function values and high-order derivatives. Our work establishes the building blocks for performing topological data analysis and visualization on implicit continuous models.

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Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.