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The optimal Pad\'{e} polynomial for reconstruction of luminosity distance based on 10-fold cross-validation

T0 review · 6 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper claims that a 10-fold cross-validation scheme selects the (2,1) Padé rational polynomial as the optimal reconstructor of luminosity distance from Pantheon+ supernova data.

desk verdict A well-intentioned but statistically underspecified attempt to make Padé order selection data-driven; the conclusion is plausible, but the procedure's reliability is not demonstrated. read the letter →

arxiv 2507.08695 v1 pith:VULVKM6P submitted 2025-07-11 astro-ph.CO

Bo Yu , Wenhu Liu , XiaoFeng Yang , Tong-Jie zhang , Yanke Tang This is my paper · ORCID
classification astro-ph.CO
keywords luminositydistancePadéapproximationcross-validationPantheon+modelselectionAkaikeInformationCriterionBayesiancosmography
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

The paper proposes a general data-driven recipe for choosing the order of a Padé rational polynomial used to reconstruct luminosity distance: partition the observed sample into folds, fit each candidate order on training folds, score held-out predictions by MSE, AIC, and BIC, average over folds, and pick the order with the best average score. Applied to the 1,701 Pantheon+ supernovae, the recipe separates four candidate orders and selects the (2,1) Padé approximation as the best explanation of the data at both low and high redshift. The central claim is that (2,1) is the optimal order among those tested, and that this cross-validation scheme is a reliable way to make such a choice for any given cosmological dataset. If the paper is right, observers gain a purely data-based alternative to the usual heuristic that numerator order should be one higher than denominator order.

What carries the argument

The machinery is the Padé rational approximation $P_{m,n}(z) = (a_0+a_1 z+\dots+a_m z^m)/(1+b_1 z+\dots+b_n z^n)$ for luminosity distance as a function of redshift, with coefficients matched to the Taylor expansion of the cosmographic distance series. Around that object the paper builds a 10-fold cross-validation model-selection pipeline: random splitting, training each candidate order on $k-1$ folds, scoring held-out predictions by MSE, AIC, and BIC, and averaging over folds. AIC and BIC penalize the number of free coefficients, which is what lets the procedure trade fit quality against polynomial complexity. The Pantheon+ supernova sample supplies the observed distance moduli that the candidate rational forms must predict.

What would settle it

Recompute the 10-fold cross-validation scores for the four candidate Padé orders with a fully specified Gaussian likelihood using the Pantheon+ covariance matrix, repeat over many random data splits, and check whether the AIC/BIC gap between (2,1) and (2,2) or (3,2) exceeds the spread of scores across splits; if it does not, the claim that (2,1) is optimal is not supported by the cross-validation evidence.

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Extended reading notes

Core claim

The central claim is that the (2,1) Padé rational polynomial is the optimal functional form for reconstructing luminosity distance from Pantheon+ data, and that this can be established by cross-validation rather than by theoretical priors. Using 2-, 5-, 10-, 20-, and 50-fold validation, the authors first screen Padé orders by mean squared error below 0.05, retain orders (2,1), (2,2), (3,1), and (3,2), and then compare fold-averaged MSE, AIC, and BIC. In 10-fold cross-validation the (2,1) order scores best on all three criteria, and MCMC fitting of its free parameters gives $q_0 = -0.52^{+0.15}_{-0.12}$, $j_0 = 1.5^{+1.1}_{-1.2}$, and $H_0 = 71.84^{+0.25}_{-0.19}$ km/s/Mpc. The paper concludes that the (2,1) Padé approximation explains Pantheon+ data at low and high redshifts and agrees with earlier theoretical arguments favoring numerator order one higher than denominator order.

Load-bearing premise

The ranking that selects (2,1) depends on AIC/BIC differences that the paper itself calls not significant, and the likelihood function and the effective number of free parameters used in those criteria are never specified.

Editorial extensions

If this is right

  • If the conclusion is correct, the (2,1) Padé form should be adopted as the default rational cosmographic reconstruction for Pantheon+ and similar supernova distance data, replacing ad hoc order choices.
  • The cross-validation procedure gives a template for order selection on other cosmological observables, such as Hubble parameter or BAO data, without assuming a specific cosmological model.
  • The fitted parameter values from the (2,1) fit provide a model-independent reference point against which parameter estimates from full cosmological model fits can be checked.
  • The result reinforces the rule that numerator order one above denominator order is preferred, because that pattern now emerges from data alone rather than from theoretical convenience.

Reading between the lines

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

  • A natural next test is whether (2,1) still wins under repeated 10-fold cross-validation with many random seeds; the paper reports no error bars on the AIC/BIC differences, so the stability of the ranking is an open question.
  • Applying the same data-driven scheme to independent distance catalogs would reveal whether (2,1) is a feature of the Pantheon+ sample specifically or of the underlying distance-redshift relation more generally.
  • If the AIC/BIC gaps are within noise, the pipeline could be extended to score candidate orders by predictive calibration, such as the coverage of validation residuals, rather than by point estimates of information criteria.
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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

6 major / 5 minor

Summary. The manuscript proposes a data-driven procedure, based on k-fold cross-validation, to select the order of a Padé polynomial used to reconstruct the luminosity distance from type-Ia supernova data. The procedure is applied to the Pantheon+ sample, and on the basis of average MSE, AIC, and BIC scores in 10-fold CV the authors conclude that the (2,1) Padé approximation is optimal and that the scheme has a 'remarkable ability to distinguish' Padé orders. The paper also reports MCMC constraints on the (2,1) model parameters.

Significance. If the statistical procedure were fully specified and validated, the paper would offer a practical, data-driven alternative to the heuristic rules previously used to choose Padé orders in cosmography, and its conclusion that (2,1) is optimal agrees with earlier work by Capozziello et al. (2020). The manuscript is honest about limitations such as dependence on the Pantheon+ dataset and on the data-splitting ratio, and it provides explicit Padé expressions. However, the paper's own evidence does not yet support the strength of the claimed conclusion: the likelihood underlying AIC/BIC is undefined, the CV scores have no uncertainties, and several numerical and consistency issues remain.

major comments (6)
  1. [Section 3.2, Eqs. (11)-(12)] The AIC and BIC values cannot be reproduced because the maximum likelihood L_max is never defined. It is not stated whether L_max is constructed from the χ2 of Eq. (1) with the full Pantheon+ covariance matrix or from a Gaussian likelihood based on the MSE of Eq. (10); the two choices lead to different AIC/BIC values. Furthermore, the number of free parameters m is ambiguous: Eq. (6) contains j0 as a coefficient, the 'where' block after Eq. (9) fixes j0 = 1, and Table 1 reports a fitted j0 = 1.5^{+1.1}_{-1.2}. Since AIC and BIC depend linearly on m, this ambiguity can change the penalty and could flip the ranking of Padé orders. The authors should state L_max explicitly and give m for each candidate order under a consistent treatment of j0.
  2. [Section 4] The conclusion that the (2,1) Padé approximation is optimal is not statistically supported by the reported scores. The authors state that 'the difference of average of BIC and AIC is not significant,' yet they select (2,1) by looking at insets that amplify those small differences. No uncertainty estimates (e.g., standard errors over the ten folds or over repeated random splits) are reported for the average MSE, AIC, or BIC, so the ranking may be an artifact of the random fold partition. Please report per-fold values, standard errors, and a paired significance test for the AIC/BIC differences, and specify the number of repeats and random seed used in Steps 1-3.
  3. [Sections 2 and 4, Step 5] The sample size used in the analysis is inconsistent. The data section states that Pantheon+ comprises '1701 light curves of 1550 unique' supernovae, while Eq. (1) and Step 5 refer to '1071 SNe Ia data' from Koussour et al. (2024). Since n enters the BIC in Eq. (12) and determines the construction of the CV folds, the paper must clarify which of these samples is used and how the number 1071 is obtained.
  4. [Equations (6)-(9) and Table 1] The Padé expressions for the luminosity distance are written as 1/H0 times a dimensionless combination of z and the cosmographic coefficients, with no factor of c. In standard units, the leading factor for d_L should be c/H0, or an explicit convention c=1 must be stated. As written, substituting H0 in km/s/Mpc gives a quantity whose units are not Mpc, so Eq. (2) would not produce the correct distance modulus. The authors should introduce c explicitly or declare the unit convention.
  5. [Section 4, Step 2] The pre-selection of candidate Padé polynomials using the threshold 'average of MSE less than 0.05' is introduced without any statistical justification. This ad hoc cutoff can remove a model with slightly higher MSE but substantially better AIC/BIC, and it is applied to an average that carries no reported uncertainty. Please justify the threshold or omit the pre-selection step and compare all orders.
  6. [Figures 2-8] The manuscript as provided does not contain the actual figures that hold the numerical evidence: Figures 2-8 appear as placeholders such as 'Figure ??' in the text, and Figures 1 and 2 share the same caption. Without these figures and the underlying numerical values, the reported CV scores and MCMC contours cannot be inspected. The figures and the exact numbers must be supplied for the central claim to be verifiable.
minor comments (5)
  1. [References] The reference to Akaike et al. (1973) in the reference list is incomplete; it should include the full title and publisher information.
  2. [Figures 1 and 2] Figure 1 and Figure 2 have identical captions; please differentiate them to indicate which shows the preliminary candidates and which shows the final fits.
  3. [Table 1] The MCMC setup (priors, number of walkers, chain length, burn-in) is not described, so the reported parameter uncertainties in Table 1 cannot be reproduced.
  4. [Throughout] The text has several typographical and grammatical errors, e.g., 'red-shifts' in the abstract and 'Sec.4 present a comparison' in Section 1; a careful language edit is needed.
  5. [Figure 8] The notation '1 δ, and 2 δ' in Figure 8 should be replaced by the standard '68% and 95% confidence regions'.

Circularity Check

0 steps flagged · score 2.0 of 10

No load-bearing circularity: the 10-fold CV order selection is data-driven and benchmarked externally; the only self-citation is non-load-bearing, and the ΛCDM-constrained Padé templates and unquantified AIC/BIC are correctness concerns, not circular reductions.

full rationale

The paper's chain is: adopt Padé templates P2,1, P2,2, P3,1, P3,2 from Capozziello et al. (2020) (Sec. 3.1, Eqs. 6-9); split Pantheon+ data into k folds; compute average validation MSE, AIC, and BIC per order; rank them; pick order (2,1); then check consistency with prior external work (Capozziello et al. 2020; Aviles et al. 2014; Petreca et al. 2024). No step in this chain makes the output equivalent to the input by construction. The order ranking is an empirical outcome: nothing in the ΛCDM coefficient hierarchy in the 'where' block forces the (2,1) rational form to win over (2,2), (3,1), or (3,2). The AIC/BIC formulas (Eqs. 11-12) are never instantiated with a stated Lmax or per-model parameter count m, so the reported scores cannot be reproduced, and the paper itself admits 'the difference of average of BIC and AIC is not significant' (Sec. 4) before selecting (2,1) from amplified insets; those are statistical-validity and reproducibility flaws, not circularity. The templates are fiducial-model-dependent (q0 = 3/2 Ωm0 − 1, j0 = 1, s0 = 1 − 9/2 Ωm0, ...), which undercuts the 'non-parametric / only based on observations' framing, but this is a model-dependence caveat: the same fiducial expansion is used for all four candidates and does not predetermine the winner. The only self-citations (Yu et al. 2020; Yu et al. 2021 in Sec. 1) are introductory remarks and are not load-bearing; methodological support comes from external citations (Allen 1974; Kohavi 1995; Zhang & Yang 2015). Hence no circular step is exhibited.

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

The analysis depends on fitted cosmological parameters (H0, q0/Ωm0, j0, and possibly s0/l0/m0), on the standard Padé construction, and on a fiducial flat ΛCDM relation for the cosmographic series. No new entities are introduced. The ad hoc MSE threshold and the nonstandard fold-averaged AIC/BIC are modeling choices specific to this paper that affect the selection result.

free parameters (4)
  • H0 = 71.84 +0.25/-0.19 km/s/Mpc (68.3% C.L., Table 1)
    Hubble constant entering all Padé luminosity distance formulas; fitted by MCMC to Pantheon+ data.
  • q0 (or Ωm0) = -0.52 +0.15/-0.12 (Table 1)
    Deceleration parameter; defined via q0 = (3/2)Ωm0 - 1 in Sec. 3.1 and fitted, or equivalently Ωm0 is fitted.
  • j0 = 1.5 +1.1/-1.2 (Table 1)
    Jerk parameter; defined as j0 = 1 in Sec. 3.1 but reported as a fitted parameter in Table 1, an unresolved inconsistency that affects the parameter count in AIC/BIC.
  • s0, l0, m0 = not reported
    Higher-order cosmographic parameters appearing in P2,2, P3,1, and P3,2; the paper expresses them as functions of Ωm0 in Sec. 3.1, so they are not independent if that relation is used, but the manuscript never states whether they are fixed or fitted in the model comparison.
assumptions (4)
  • standard math Padé approximants are constructed from the Taylor expansion of the luminosity distance using the standard coefficient-matching conditions (Eqs. 3-5).
    The Padé construction is a standard mathematical technique; the specific luminosity distance coefficients are taken from Capozziello et al. (2020) rather than derived in this paper.
  • domain assumption The cosmographic parameters are linked to a flat ΛCDM model through q0 = (3/2)Ωm0 - 1, j0 = 1, s0 = 1 - (9/2)Ωm0, etc. (Sec. 3.1).
    This relation restricts the Padé templates to a specific fiducial cosmology, so the 'reconstruction' is not fully model-independent, yet the paper presents it as non-parametric reconstruction.
  • ad hoc to paper Candidate Padé orders are pre-selected using an ad hoc threshold that the average MSE must be less than 0.05 (Sec. 4, Step 2).
    This threshold has no statistical justification and could bias the subsequent comparison by excluding orders that perform poorly on one criterion before AIC/BIC are applied.
  • ad hoc to paper AIC and BIC are computed as fold-averaged values over cross-validation folds (Sec. 3.3).
    Standard AIC/BIC are computed on the full dataset maximum likelihood; averaging them over validation folds is a nonstandard implementation that the paper does not justify or compare with the standard definition.

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Pith. "Pith review of The optimal Pad\'{e} polynomial for reconstruction of luminosity distance based on 10-fold cross-validation." pith.science (2026). https://pith.science/paper/VULVKM6P

@misc{pith2026250708695,
  author       = {Pith},
  title        = {Pith review of: The optimal Pad\'e polynomial for reconstruction of luminosity distance based on 10-fold cross-validation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VULVKM6P}},
  note         = {Machine review of arXiv:2507.08695}
}
read the original abstract

The cosmography known as the Pad\'{e} polynomials has been widely used in the reconstruction of luminosity distance, and the orders of Pad\'{e} polynomials influence the reconstructed result derived from Pad\'{e} approximation. In this paper, we present a more general scheme of selecting optimal Pad\'{e} polynomial for reconstruction of luminosity distance based on 10-fold cross-validation. Then the proposed scheme is applied to Pantheon+ dataset. The numerical results clearly indicate that the proposed procedure has a remarkable ability to distinguish Pad\'{e} approximations with different orders for the reconstruction of the luminosity distance. We conclude that the (2,1) Pad\'e approximation is the optimal approach that can well explain Pantheon+ data at low and high red-shifts. Future applications of this scheme could help choose the optimal model that is more suitable for cosmological observation data at hand and gain a deeper understanding of the universe.

Figures

Figures reproduced from arXiv: 2507.08695 by the authors.

Figure 1
Figure 1. Hubble diagram for Pantheon+ Sample. The fitting results for distance modulus of preliminary candidate Pad´e polynomials to each other, which is similar to Aviles et al. (2014). From the perspective of the constructions of Pad´e se￾ries and the mathematical rules derived from the de￾generacy among coefficients, Capozziello et al. (2020) singled out the most suitable rational approximation is the one that minimizes t… view at source ↗
Figure 2
Figure 2. Hubble diagram for Pantheon+ Sample. The fitting results for distance modulus of preliminary candidate Pad´e polynomials REFERENCES Akaike, H., Petrov, B. N., & Czaki, F. 1973, 2nd Int.Sympo.on Information Theory Allen, D. M. 1974, Technometrics, 16, 125 Arlot, S., & Celisse, A. 2010, Statistics Surveys, 4, 40 , doi: 10.1214/09-SS054 Aviles, A., Bravetti, A., Capozziello, S., & Luongo, O. 2014, Phys. Rev. D, 90, 043… view at source ↗
Figure 3
Figure 3. The average of BIC of preliminary candidate Pad´e polynomials in 2,5,10,20 and 50 fold CV, respectively. The difference of average of BIC in 10-fold CV that is amplified is shown in inset. Riess, A. G., et al. 1998, The Astronomical Journal, 116, 1009, doi: 10.1086/300499 Schwarz, G. 1978, The Annals of Statistics, 6, 461 Scolnic, D. M., et al. 2018, The Astrophysical Journal, 859, 101, doi: 10.3847/1538-4357/aab9bb… view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: The average of AIC of preliminary candidate Pad´e polynomials in 2,5,10,20 and 50 fold CV, respectively. The difference of average of AIC in 10-fold CV that is amplified is shown in inset [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]
Figure 5
Figure 5. Figure 5: The MSE of preliminary candidate Pad´e polynomials in 10-fold CV [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]
Figure 6
Figure 6. Figure 6: The AIC of preliminary candidate Pad´e polynomials in 10-fold CV [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]
Figure 7
Figure 7. Figure 7: The BIC of preliminary candidate Pad´e polynomials in 10-fold CV [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]
Figure 8
Figure 8. Figure 8: The contour for 1δ, and 2δ in the two-dimensional parameter spaces of P(2,1), and the marginal distribution function in one-dimensional parameter space [PITH_FULL_IMAGE:figures/full_fig_p013_8.png]

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