Data-driven Approach for Interpolation of Sparse Data

B. McKinnon; D.G. Ireland; R.F. Ferguson

REVIEW 6 major objections 6 minor 1 cited by

Data-driven Approach for Interpolation of Sparse Data

T0 review · 6 major / 6 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read Gaussian processes can interpolate sparse hadron data with quantified uncertainties, removing the need for arbitrary weighting.

desk verdict A useful applied-GP recipe for sparse hadron data, but the calibration evidence is circular and the headline robustness claim outruns the validation. read the letter →

arxiv 2505.01473 v1 pith:6JAJYK2F submitted 2025-05-02 physics.data-an nucl-exstat.ML

R.F. Ferguson , D.G. Ireland , B. McKinnon This is my paper · ORCID

classification physics.data-annucl-exstat.ML

keywords GaussianprocessessparsedatainterpolationhadronspectroscopyhyperparameterselectioncoveragelossuncertaintyquantificationconsistencyBayesianinference

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

This paper argues that Gaussian process regression, with length scales chosen by a coverage-based loss rather than the standard marginal likelihood, can build smooth interpolated datasets from the sparse, unevenly binned measurements typical of hadron resonance studies. The point is to replace arbitrary weighting of different experiments in model fits with predictions that carry their own quantified uncertainties. If the claim holds, data from many experiments can be combined on equal footing, and inconsistencies between datasets can be flagged before a theoretical fit is performed. The authors test the method on pseudodata built from Legendre polynomials plus Gaussians and on real photoproduction and kaon charge-exchange data.

What carries the argument

The engine is a Gaussian process with an RBF kernel $\kappa(\vec a,\vec b,\vec l)=\exp(-\sum_k \|a_k-b_k\|^2/(2 l_k^2))$, whose prediction for an unmeasured point is the conditional mean $\vec\mu_* = K_*^T K^{-1}\vec y$ and covariance $\Sigma_* = K_{**}-K_*^T K^{-1} K_*$. Because the log marginal likelihood of Eq. (1) is unstable for sparse data, the authors introduce the coverage-loss function $f(\vec l)=\int_0^3 |M(c\sigma,\vec l)-P(c\sigma)|\,dc$ of Eq. (2), which scores how well the GP's error bands contain the expected fraction of points. The loss is minimised with a standard minimiser in 1D and, in higher dimensions, by MCMC sampling followed by K-means clustering with silhouette scoring to pick among the loss surface's multiple modes; predictions are restricted to the convex hull of the measured points, with bin edges added when binning information is available.

What would settle it

Take a sparse pseudodataset with a known generating function, compute the coverage-loss-optimal length scale and the marginal-likelihood-optimal length scale, and compare predictive mean squared error and empirical coverage on out-of-sample points; if the coverage-loss choice is no better than the marginal likelihood and its uncertainty bands undercover on held-out points, the central claim fails. The paper's Table I already measures in-sample pull, so a fully independent check requires points not used in the GP fit.

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Extended reading notes

Core claim

The central claim is that a Gaussian process whose radial-basis-function length scales are optimised against a coverage loss—the integrated absolute difference between the observed and expected fractions of training points inside $0$ to $3\sigma$ bands of the GP mean—interpolates sparse hadron-spectroscopy datasets as accurately as the original data themselves, with well-quantified uncertainties. The paper demonstrates this by showing that pull distributions of sampled GP points match statistical expectations and that fits to the known functional form using GP-sampled points reproduce fits to the original pseudodata. It further claims the same machinery can expose inconsistencies between experiments, as illustrated by a double-ridged log-probability surface for $K^-p\to \bar{K}^0 n$ world data, where a least-squares fit would sit between the ridges and miss the maximum likelihood.

Load-bearing premise

The load-bearing premise is that the coverage-based loss computed on the training data is a valid replacement for the marginal likelihood in choosing length scales, and that in-sample coverage checks plus GP-sampled points are enough to certify the interpolated values and uncertainties.

Editorial extensions

If this is right

Fits of coupled-channels models can use GP-interpolated points with the same kinematic density for every observable, removing experiment-specific relative weights.
A combined probability surface built from individual GP fits can replace a single weighted average when datasets disagree, preventing least-squares fits from landing in low-probability valleys.
The method supplies a value and uncertainty at any kinematic point inside the measured domain, enabling model tests in kinematic regions where no direct measurement exists.
The same GP treatment can be applied to any observable of a reaction, so all measured quantities enter a theoretical fit in the same way.

Reading between the lines

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

The coverage loss is a frequentist calibration criterion rather than a proper scoring rule; for very small samples the step-like 1$\sigma$ curve in Fig. 1 suggests the minimum may be sensitive to the discrete placement of points, so a bootstrap or repeated-pseudodata study of length-scale stability would be a natural extension.
Because the GP uncertainty is itself an estimate, any downstream fit that treats GP-sampled points as independent data will understate the total uncertainty; a two-stage analysis should propagate the GP covariance or use the full posterior predictive distribution.
The consistency-surface idea could be turned into a quantitative disagreement measure, such as overlap integrals between experiment-specific GP posteriors, and applied to other multi-experiment fields like lattice QCD averages or nuclear reaction databases.
If the coverage loss were replaced by a proper scoring rule evaluated on a small validation set, the method might combine the stability of the coverage criterion with the statistical guarantees of the marginal likelihood.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague,

The useful thing here is a practical recipe: use GPs to interpolate sparse hadron-spectroscopy datasets and return a smooth surface with pointwise uncertainties, then use those surfaces to compare experiments. The core GP equations are textbook; the new bit is the coverage-based loss for choosing length scales when marginal likelihood gives pathological results on sparse data, and a mixture “probability surface” for flagging inconsistent experiments. That is a genuine adaptation, not a new framework, and the code is public. The real-data examples (CLAS Σ, K−p→Kbar0n world data) show the method can produce plausible surfaces and expose scatter between experiments. On those grounds the paper deserves a serious referee.

The soft spot is the validation, and it is central. Eq. (2) defines the length-scale loss as the integrated coverage deficit computed on the training points. Table I then reports coverage of “Known Points” as evidence that the GP uncertainties are calibrated. That is the same statistic you optimized, so it cannot validate anything. The “GP Points” column is closer to an out-of-sample check only in the sense that the points are drawn from the GP posterior, not from new data; the GP is both the source of the test points and the model being tested, and the pseudodata are all smooth Legendre-Gaussian surfaces, so narrow resonance-like features never get tested. The hyperparameter search is run on f(l), not on a posterior, so calling it Bayesian inference and using the corner plots to quote uncertainty on length scales is not justified. The consistency surface in Sec. VI is also a sum of normalized Gaussians from independent GP fits, not a product of likelihoods; it might be a useful diagnostic, but it is not a probability of the cross section given the experiments, so the “maximum likelihood” language overreaches.

None of this kills the paper. The interpolation tool is plausible, the coverage loss is a reasonable heuristic for sparse data, and a head-to-head with marginal-likelihood GP plus held-out validation would settle the main question. The authors need to add that before the robustness claim in the abstract can stand.

For peer review: yes, send it out. It is exactly the kind of paper a referee can improve with concrete demands. For me personally, I would not cite it yet; after a revision with honest held-out tests, I might.

Referee Report

6 major / 6 minor

Summary. This paper proposes a Gaussian-process (GP) interpolation method intended to replace arbitrary weighting of sparse hadron spectroscopy data. The proposed model uses a squared-exponential (RBF) kernel, and the length-scale hyperparameters are selected by minimizing a coverage-loss function f(l) defined in Eq. (2), which compares the empirical fraction of training points inside c-sigma bands with the Gaussian expectation; the minimization is performed with MCMC and K-means clustering. The authors validate the method on 100 pseudodata surfaces made of Legendre polynomials times Gaussian energy functions (Eq. (4)), then apply it to CLAS photon-beam asymmetry data and to seven experiments for K-p -> Kbar0 n, constructing a combined probability surface to assess dataset consistency.

Significance. If the validation were trustworthy, the method would be a convenient and transferable recipe for hadron spectroscopy analyses, and the manuscript is generally clear and backed by a public code repository. However, the evidence as presented does not support the central robustness claim: the hyperparameter-selection criterion and the headline calibration statistic are the same quantity, the 'GP datapoints' are not an independent test, the pseudodata are unrealistically smooth, and the MCMC procedure is not Bayesian in any standard sense. With an independent held-out validation and a defensible hyperparameter-selection scheme, the tool could be of practical value to the hadron-spectroscopy community.

major comments (6)

[III B and Table I] The coverage-loss function f(l) in Eq. (2) and the validation statistic in Table I are the same object. The 'Known Points' column reports M(csigma,l) after l is chosen to minimize |M(csigma,l)-P(csigma)|, so the displayed agreement is enforced by construction. The in-sample coverage numbers therefore do not constitute evidence of calibrated GP uncertainties; a held-out validation set or a cross-validation scheme is required.
[III B, Appendix B] The manuscript repeatedly calls the procedure 'Bayesian inference' (Abstract, Section III B), but f(l) is an ad hoc loss, not a likelihood or posterior, so running MCMC on it does not yield posterior samples or hyperparameter uncertainties. In addition, Appendix B sets T_w=0 for the integrated autocorrelation time, which makes tau_w=1/2 and tau_stability=0 for every chain; the stated autocorrelation convergence criterion is thus vacuous. The convergence assessment therefore rests only on the Gelman-Rubin statistic, and the claim of Bayesian hyperparameter optimization is unsupported.
[IV A and Table I] The 'GP Points' column does not validate interpolation at unobserved kinematics. These points are drawn from the GP predictive distribution conditioned on the same training data, so they are not independent of the hyperparameter choice and cannot reveal miscalibration away from the training points. The text's assertion that the GP can use the full dataset 'without arbitrary splitting' does not remove the need for an external validation; leave-one-energy-out or a separate test surface would be a minimal fix.
[Table II] Table II does not support the stated conclusion. For the GP datapoints the pull variances include mu0=0.51, c2=0.56, mu2=0.50, and sigma2_3=0.68; even the pseudodata column has mu2=0.41 and mu1=0.55. Since a correctly calibrated pull distribution has variance 1, these values indicate overestimated fit uncertainties or neglected correlations. The claim that both columns are 'consistent with statistical expectations' is contradicted by the reported numbers.
[IV, Eq. (4)] The validation family in Eq. (4) consists of four low-order Legendre polynomials times Gaussian energy functions, which are smooth and broad. Typical hadron resonance datasets contain narrow peaks, threshold effects, and rapid angular structure; the abstract's claim that GPs are robust for 'typical datasets used in hadron resonance studies' is not established by tests on this unrealistically smooth family. Adding sharp-featured test functions is essential to support the generality claim.
[VI] The combined probability surface is not a proper likelihood. Summing normalized Gaussian GP predictive densities from independent experiments and renormalizing at each energy treats the experiments as independent likelihood terms with no common-systematic component and ignores correlations within each GP fit. The plot in Figure 8 may be a useful heuristic diagnostic, but the claim that a least-squares fit would produce 'misleading inferences' because it is not maximum likelihood requires this surface to be a valid likelihood, which is not shown.

minor comments (6)

[Eq. (2)] The notation '3X c>0' for the discrete approximation of the integral is non-standard; the sum over a grid of c values in (0,3] should be written explicitly.
[III B] The phrase 'predicated percentage' should be 'predicted percentage'.
[II] The covariance matrix is defined inconsistently: first as K = kappa(X,X) + vector e^2 I_n and then correctly as K_ab = kappa(x_a,x_b) + delta_ab e_a^2; the vector notation should be removed or defined.
[Table I] The measured percentages are quoted without statistical uncertainties; with 100 pseudodata surfaces and variable numbers of points, the sampling error on each percentage should be reported.
[III C] The angle-bin-width formula uses symbols m and a_m without definition; the indexing for degenerate datapoints should be stated explicitly.
[Reference [3]] The GitHub repository is cited without a version or commit identifier; for reproducibility, please include the commit hash or an archived DOI.

Circularity Check

1 steps flagged · score 6.0 of 10

In-sample coverage validation restates the fitted loss; remaining pseudodata checks are only partly independent, so partial circularity.

self definitional [Section III B (Eq. 2) and Section IV A (Table I)]
"f(⃗l) = ∫_0^3 |M(cσ,⃗l)−P(cσ)|dc ≈ Σ_{c>0} |M(cσ,⃗l)−P(cσ)|, (2) ... M(cσ,⃗l) is the measured percentage of datapoints falling within cσ of the GP mean for a given length scale ⃗l ... Table I: Different σ bands and the measured percentage of points contained in each ... Measured Percentage of Known Points (%) 51.2 68.1 94.8"

The length scale is selected by minimizing f(⃗l), which is literally the integrated absolute difference between the measured coverage M(cσ,⃗l) of the known training points and the nominal Gaussian coverage P(cσ). Table I's 'Known Points' column reports exactly this same M(cσ,⃗l) at the selected length scale (51.2%, 68.1%, 94.8% against the expected 50%, 68.3%, 95%). The in-sample agreement is therefore imposed by the selection objective: it restates the fitted target rather than independently confirming that the GP predictive variances are calibrated. The 'GP Points' column is a separate check, but it is conditional on the length scale already chosen from the training coverage and on smooth pseudodata generated from Eq.

full rationale

The central methodological novelty is Eq. (2), a coverage-deficit loss used to choose GP length scales and to support the claim of robust, quantified uncertainties. The validation in Table I has a 'Known Points' column that is the same statistic the loss minimizes, making that part of the validation self-definitional. The 'GP Points' column and the coefficient-pull tests in Section IV B provide some independent evidence that interpolation on smooth Legendre-Gaussian pseudodata preserves the underlying functional form, and the real-data applications are demonstrations rather than tests of the method. There is no load-bearing self-citation chain: the citations to Rasmussen and Williams and to standard MCMC packages are external support. However, because the headline robustness and uncertainty-quantification claim rests in part on the circular known-points coverage check, and because the ad hoc loss is introduced without derivation from a likelihood, the paper is partially circular rather than fully self-contained.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

No new physical particles, forces, or conserved quantities are introduced. The main invented components are statistical constructs: the coverage-loss objective and the mixture-based probability surface; these are captured as axioms and free parameters above.

free parameters (3)

RBF length scale l_k (per kinematic dimension) = not reported; selected by minimizing Eq. (2)
The GP predictions depend on the posterior whose covariance is set by l_k; the paper tunes them by minimizing the coverage loss f(l), so the central result is conditional on these fitted values.
Integration range c in (0,3] = c_max = 3
The loss integrates absolute coverage deviations over 0 to 3 sigma; this upper bound is chosen by hand, not derived.
MCMC convergence thresholds = Rhat < 1.18; tau_stability < 0.15
The paper uses looser convergence thresholds than the usual Rhat < 1.1 and tau < 0.1 to save computation; the resulting posterior samples and selected length scales depend on these thresholds.

assumptions (6)

domain assumption The RBF kernel is a valid covariance model for the hadron observables being interpolated.
Section III A uses only the RBF kernel with no comparison to other kernels or check that the true observable surfaces have this smoothness.
ad hoc to paper The fraction of data points within c sigma of the GP mean should match the Gaussian expected fraction, and minimizing this mismatch is a valid hyperparameter selection rule.
Section III B defines the loss in Eq. (2) without deriving it from a likelihood or posterior; the same statistic is later used as validation in Table I.
ad hoc to paper A GP can be validated using the same data used for hyperparameter optimization, without a held-out split.
Section IV validates with the training pseudodata and with points sampled from the GP itself; no held-out test locations are used to assess predictive accuracy.
ad hoc to paper The sum of normalized Gaussian predictions from different experiments is a valid combined probability surface.
Section VI constructs the surface by summing Gaussians; for independent experiments the combined likelihood is a product, so this is an unstated and questionable modeling choice.
domain assumption The quoted measurement uncertainties are independent, Gaussian, and accurate.
Section II treats y_i +- e_i as independent Gaussian noise with known e_i; no account is taken of correlated or systematic uncertainties.
standard math Standard Gaussian process and multivariate Gaussian conditioning results are valid.
Section II invokes Rasmussen and Williams [1] and Murphy [2, Theorem 4.3.1] for the conditional formulas; these are textbook results.

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Cite this review

Pith. "Pith review of Data-driven Approach for Interpolation of Sparse Data." pith.science (2026). https://pith.science/paper/6JAJYK2F

@misc{pith2026250501473,
  author       = {Pith},
  title        = {Pith review of: Data-driven Approach for Interpolation of Sparse Data},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6JAJYK2F}},
  note         = {Machine review of arXiv:2505.01473}
}

read the original abstract

Studies of hadron resonances and their properties are limited by the accuracy and consistency of measured datasets, which can originate from many different experiments. We have used Gaussian Processes (GP) to build interpolated datasets, including quantification of uncertainties, so that data from different sources can be used in model fitting without the need for arbitrary weighting. GPs predict values and uncertainties of observables at any kinematic point. Bayesian inference is used to optimise the hyperparameters of the GP model. We demonstrate that the GP successfully interpolates data with quantified uncertainties by comparison with generated pseudodata. We also show that this methodology can be used to investigate the consistency of data from different sources. GPs provide a robust, model-independent method for interpolating typical datasets used in hadron resonance studies, removing the limitations of arbitrary weighting in sparse datasets.

Figures

Figures reproduced from arXiv: 2505.01473 by the authors.

**Figure 1.** and are of the form: f( ⃗l) = n|M(cσ,⃗l) − P(cσ)| where • M(cσ,⃗l) is the measured percentage of datapoints falling within cσ of the GP mean for a given length scale ⃗l, • P(cσ) is the corresponding predicated percentage. • n is the total number of datapoints [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗

**Figure 2.** An example loss function distribution. For observables dependent on one kinematic variable, it is sufficient to use a standard minimiser to find the optimal length scale. However, increasing the kinematic dimensions, to even 2D, produces a multi-modal surface for the loss function; to solve this, a Markov Chain Monte Carlo (MCMC) can be used. In the MCMC procedure outlined in this paper, the chain is run until two d… view at source ↗

**Figure 3.** An example 2D corner plot showing the mul [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: 1D projections of fits from a generated 2D [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

**Figure 6.** Figure 6: 1D projections of GP fit of Σ The usefulness of this method is that all measured observables required for a fit by a theoretical model can be treated in the same way; the GP method can return a value and uncertainty for an observable at any point as a function of kine…

**Figure 7.** Figure 7: Individual GP fits on experiments for K−p → K¯ 0n. This is a case where the standard procedure of combining estimates and uncertainties to give a single estimate as a function of energy might lead to misleading results. We can use the results of the GP fits to constru…

**Figure 8.** Figure 8: Log probability surface for K−p → K¯ 0n. It is clear from the plot that there is significant nontrivial structure in the likelihood surface, with a clear double ridged structure at lower energies. A theoretical fit that minimises least squares will attempt to move to …

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Forward citations

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Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

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arXiv 2025

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