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Generalized Parton Distributions from Symbolic Regression

T0 review · 4 major / 6 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read This paper claims that lattice QCD data for the isovector GPD $H_{u-d}(x,t,\xi=0)$ approximately factorize in the measured kinematic region, with any factorization breaking concentrated at low $x$.

desk verdict New method (ECC) is worth a close look, but the low-x localization of factorization breaking is not supported by the moment-ratio evidence and should be dialed back in revision. read the letter →

arxiv 2504.13289 v3 pith:NVUGLWKF submitted 2025-04-17 hep-ph hep-lat

Andrew Dotson , Zaki Panjsheeri , Anusha Reddy Singireddy , Douglas Q. Adams , Emmanuel Ortiz-Pacheco , Marija Cuic , Yaohang Li , Huey-Wen Lin
show 6 more authors
Simonetta Liuti Matthew D. Sievert Marie Boer Gia-Wei Chern Michael Engelhardt Gary R. Goldstein
This is my paper · ORCID
classification hep-phhep-lat
keywords generalizedpartondistributionssymbolicregressionlatticeQCDfactorizationtransversespatialdensitiesexpansioncoefficientclusteringDiracformfactorprotontomography
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 tries to establish that symbolic regression can turn a small 13-by-5 grid of lattice QCD points for the quark GPD $H_{u-d}(x,t,\xi=0)$ into interpretable analytic expressions, and that those expressions reveal an approximate separation of $x$ and $t$ dependence in the trained region $0.255\le x\le 0.855$, $|t|\le 1\,\mathrm{GeV}^2$. It matters because the Fourier transform of the GPD in $t$ gives the quark's transverse spatial distribution: a factorized GPD means the quark transverse radius is nearly independent of $x$, whereas QCD-based intuition expects high-$x$ quarks to be more point-like. The paper argues that the apparent factorization breaking seen in integrated lattice moment ratios is not contradictory, because it can be entirely focused in the low-$x$ region outside current data. A reader should care because the paper offers a concrete route to extracting proton tomography from sparse lattice data while quantifying how much of the result is assumption rather than data.

What carries the argument

The load-bearing machinery is Expansion Coefficient Clustering: each of roughly a thousand independent symbolic-regression replicas is expanded in Taylor polynomials around the midpoint of the training region, $x=0.555$, and the resulting coefficient vectors are clustered so that replicas with the same extrapolation behavior form bands. This is paired with two custom training criteria: a force-factorization penalty that nudges replicas toward the form $f_1(x)f_2(t)$, and a semi-Reggeized form $x^\alpha(1-x)^\beta P(x)g(t)$ imposed through a redundant-variable trick. A finite-moment filter keeps only replicas whose integral over $x$ at $t=0$ is well defined, which both removes poles and, without being enforced, yields a Dirac form factor $A_{10}(0)$ near the expected value of 1.

What would settle it

Obtain lattice QCD points for $H_{u-d}(x,t,\xi=0)$ at $x<0.255$ for the same $|t|$ values up to about $1\,\mathrm{GeV}^2$. If the fixed-$t$ ratio $R(x,|t|)=H_{u-d}(x,0)/H_{u-d}(x,|t|)$ remains within about 10 percent of constant in that new low-$x$ region, or if it deviates by substantially more than 10 percent within the originally trained $0.255\le x\le 0.855$ range, the paper's claim that factorization breaking is confined to low $x$ is falsified.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that unconstrained symbolic-regression fits to the lattice data still approximately factorize numerically: the fixed-$t$ ratio $R(x,|t|)=H_{u-d}(x,0)/H_{u-d}(x,|t|)$ stays within about 10 percent of constant in the trained region, and the moment ratios $A_{20}(t)/A_{10}(t)$ and $A_{30}(t)/A_{10}(t)$ reconstructed from the fitted replicas are nearly $t$-independent. The paper interprets this as evidence that the lattice data allow factorization breaking only in the low-$x$ region, and that this is compatible with existing integrated lattice moment results. It also claims a methodological discovery: comparing the distributions of best-fit versus forced-factorized mean-squared errors, through a Kullback-Leibler divergence, separates factorizing sources from non-factorizing ones, showing that symbolic regression can act as a hypothesis-testing tool rather than just a fitting tool.

Load-bearing premise

The symbolic expressions trained on the 13-by-5 grid are treated as reliable predictions outside that grid, especially below $x=0.255$ and above $|t|=1\,\mathrm{GeV}^2$, when computing the spatial densities, radii, and form-factor extrapolation.

Editorial extensions

If this is right

  • If the factorization claim survives, the quark transverse RMS radius $b_{\mathrm{RMS}}(x)$ is approximately constant across $0.255\le x\le 0.855$, meaning current lattice data do not yet resolve the expected shrinking of high-$x$ quark configurations.
  • The observed $t$-dependence of integrated moment ratios does not refute approximate local factorization; the tension dissolves if the breaking is confined to $x<0.255$ and $|t|\ge 1\,\mathrm{GeV}^2$.
  • Comparing best-fit and forced-factorized loss distributions gives a quantitative, reproducible test of whether any future GPD dataset is compatible with factorization.
  • The Taylor-coefficient clustering criterion offers a general convergence check for symbolic regression on small physics datasets, beyond this particular GPD extraction.
  • A finite-moment filter alone eliminates pole instabilities and biases replicas toward the correct charge sum rule, suggesting physics constraints can be injected cheaply into symbolic regression workflows.

Reading between the lines

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

  • Going beyond the paper, one could test whether the factorization pattern is special to the isovector unpolarized GPD: applying the same pipeline to the gluon GPD or the helicity-flip GPD $E$ would show whether constant radii are a general lattice-data feature or an artifact of this one observable.
  • The clusters that diverge at low $x$ imply a concrete prediction that future data can settle: once lattice calculations reach $x\approx 0.1$, the transverse radius should either grow steeply, confirming low-$x$ factorization breaking, or remain flat, contradicting the paper's low-$x$ interpretation.
  • The ECC Taylor basis centered at $x=0.555$ may hide endpoint behavior; an expansion basis better adapted to $x\to 0$ or $x\to 1$, such as a Gegenbauer or Bernoulli basis, would reveal whether the cluster structure is an artifact of the midpoint expansion.
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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

4 major / 6 minor

Summary. The manuscript applies the symbolic regression package PySR to the lattice QCD results for the isovector GPD H_{u-d}(x,t,ξ=0) from Ref. [29], with the goal of extracting interpretable analytic expressions and using them to infer physics. The authors introduce an 'Expansion Coefficient Clustering' (ECC) criterion to assess the consistency of the many symbolic solutions produced by independent SR runs. They compare three selection criteria: unconstrained Best-Fit (BF), Force-Factorized (FF), and a Semi-Reggeized form. They test the factorization of x and t dependence using fixed-t ratios, MSE comparisons between BF and FF fits, KL divergences of the MSE distributions, and moment ratios A20/A10, A30/A10. The paper claims that, in the trained region (0.255≤x≤0.855, |t|≤1 GeV²), the LQCD data approximately factorize, and that any factorization breaking is confined to low x. It then uses the fitted forms to compute transverse densities ρ(x,b_T) and the average transverse radius b_RMS(x), and compares these with phenomenological models. The central physics conclusions are that LQCD data favor approximate factorization in the measured region, and that the extracted radii show model-dependent trends in the extrapolation region.

Significance. The paper is a serious methodological contribution: it brings symbolic regression with systematic replica analysis to GPD phenomenology, and it proposes a concrete criterion (ECC) for assessing the consistency of SR solutions. The inventory of hyperparameters in Appendix B and the explicit discussion of unaccounted LQCD correlations are useful for reproducibility. If the factorization claims survive scrutiny, the result that the LQCD isovector GPD approximately factorizes in the measured x,t region would be a notable input to GPD phenomenology and to the design of future extractions. The extrapolated densities and radii are potentially interesting, but their credibility hinges on the validity of extrapolation far outside the training grid, which is not established by the presented evidence. Overall, the in-region factorization claim is defensible, but the paper's headline conclusions about low-x localization and the spatial radii overreach what the current tests demonstrate.

major comments (4)
  1. [Sec. IV C] The conclusion that 'LQCD data allow for a breaking of factorization in the x,t behavior of GPDs that is entirely focused in the low x region' is not supported by the analysis presented. The evidence cited for this localization is the t-dependence of the integrated moment ratios A20/A10 and A30/A10 in Fig. 17. These are integrals over the full x range, and A30 weights large x more heavily than small x. A non-constant A30/A10 could equally arise from factorization breaking at moderate or high x, or even from effects at |t|>1 GeV², since the integrals receive contributions from all x and t. The paper performs no test that isolates the x location of the breaking, so the phrase 'entirely focused in the low x region' is not a demonstrated result. This matters because the subsequent interpretation of b_RMS(x) and the comparison with Reggeized GGL/GK models rest on this localization. I recommend either removing the localization claim or adding a direct x-resolved test, such as evaluating the fixed-t ratio R(x,|t|) separately in x bins and testing whether deviations from unity occur only for x<0.255.
  2. [Sec. IV D] The extraction of the transverse densities ρ(x,b_T) and the average radii b_RMS(x) treats the symbolic expressions as reliable predictions outside the training region, particularly at x<0.255 and |t|>1 GeV². This is an extrapolation from a 13×5 lattice grid, and the ECC clusters diverge precisely in this extrapolation region, as shown in Figs. 6 and 10. The claim that 'all four of these distributions are a prediction of the combined SR & LQCD framework' (Fig. 20 caption) overstates the status of these curves: they are model-dependent extrapolations with no physical constraints (e.g., positivity, matching to PDFs at x→0 and x→1, or Regge behavior at small x) imposed in the BF fits. In particular, the upward turn of BF Cluster 3 as x→1 would, if taken at face value, conflict with the QCD expectation of point-like configurations, but this behavior occurs in the region where no data exist and where the different clusters disagree. I recommend reframing these as illustrative extrapolations, or adding validation checks such as training on a subset of x and testing on the held-out region, or imposing physical endpoint constraints in the loss function.
  3. [Sec. IV C] The KL-divergence comparison KL(MSE_BF||MSE_FF) lacks a BF-versus-BF baseline. The authors themselves note (Sec. IV C) that 'we are unable to establish hierarchy of source factorization breaking' and propose generating two sets of BF replicas as a baseline in future work. Without this baseline, the numerical values in Table II cannot be interpreted as an absolute measure of compatibility with factorization; a large KL value could also arise from differences in the convergence of BF versus FF runs that are unrelated to the factorization hypothesis. The paper does correctly use the overlapping histograms in Fig. 16 to support the in-region factorization claim for VGG and LQCD, and the strong separation for GGL and GK. My concern is only with the quantitative use of the KL values themselves; I recommend presenting the KL numbers as qualitative indicators and adding the BF-baseline comparison before drawing any conclusion about the relative degree of factorization breaking.
  4. [Sec. IV A 4] The treatment of uncertainties is insufficient for the strength of the factorization claim. The paper states that 'the LQCD errors are highly correlated but unaccounted for' and that the MSE and WMSE 'do not carry statistical significance.' Yet the conclusion that LQCD factorizes 'to within ≈10%' in the training region is stated without an uncertainty on that percentage. The 10% figure is a scatter over the fixed-t ratio points, not a confidence interval. Since the central physics claim is a quantitative statement about the degree of factorization, the paper should either propagate the LQCD point-to-point uncertainties into the R(x,|t|) ratios, or explicitly state that the 10% is a measure of deviation from a constant in the plotted points and is not statistically quantified.
minor comments (6)
  1. [Eq. (26)] The ratio in Eq. (26) is written as H_{u-d}(x,t)/H_{u-d}(x,t), which is identically 1; the denominator should involve a different t value, presumably t_j. Please fix the typo.
  2. [Eq. (33)] The definition of the dimensionless variable t≡−t/Λ² combined with the expression e^{−t} in Eq. (33) appears to give a t-dependence that grows with |t|, which would be unphysical for a form factor or GPD. Please check the sign convention and ensure the expression actually decreases with |t|.
  3. [Sec. IV A 3] The 'Semi-Reggeized' exemplar is selected as 'the first SR result obtained' with the required form, not as a best-fit or an ensemble average. This makes the comparison with the BF and FF exemplars in Table I and Fig. 14 potentially biased. Please clarify that this is a single, non-representative example, or run multiple RCA searches and report the distribution.
  4. [Sec. III C] The phrase 'systematic convergence' overstates what ECC demonstrates. ECC clusters replicas into families based on Taylor coefficients; it does not show convergence to a unique symbolic form. Please either rename the criterion (e.g., 'systematic consistency') or qualify that the clusters represent distinct but equally valid solutions from the SR search.
  5. [Sec. IV B] The sentence 'In Fig. I, we present a typical result for each cluster' appears to refer to Table I, which lists the functional forms, not to a figure. Please correct the cross-reference.
  6. [Throughout] There are several typographical issues, including 'Relevent', 'farther develops', 'It has been convergence is not particularly well-defined', and inconsistent use of 't' for both the physical and dimensionless variable. A careful proofreading pass is needed.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the factorization conclusion is an empirical comparison of constrained and unconstrained symbolic-regression fits, not a derivation from the hypothesis itself.

full rationale

The paper is a regression and model-comparison study rather than a closed-form derivation. The central factorization claim is tested by comparing Best-Fit (BF) symbolic-regression models, which are unconstrained in functional form, against Force-Factorized (FF) models whose custom loss in Eq. (28) forces factorized x/t dependence. Because the BF expressions are not required to factorize, and the target LQCD data are external inputs from Ref. [29], the finding that BF replicas “approximately numerically factorize” is an empirical result obtained from the fits rather than a tautology. The VGG model is factorizing by construction and is used only as a control baseline; the non-factorizing GGL and GK models provide independent falsification benchmarks, so the FF response test is not self-referential. The ECC clustering in Sec. III C groups replicas by Taylor-expansion coefficients as a convergence diagnostic; it does not insert factorization or the low-x-locality conclusion into the inputs. The extrapolated densities and radii in Figs. 18–20 are labeled as predictions of the combined SR & LQCD framework, which is an honest description of a fitted-model extrapolation, not a claim that they were derived from an independent first-principles calculation. The moment-ratio discussion in Sec. IV C may overreach in locating factorization breaking entirely at low x, since the full-x integrals A20/A10 and A30/A10 do not localize the breaking in x; however, that is a statistical-inference and robustness concern, not a circularity in which an output is equivalent to an input by construction. No load-bearing step relies on a uniqueness theorem or on self-citations to forbid alternative interpretations, and the cited lattice data and benchmark models are independent inputs. Therefore no circular step meets the evidentiary bar of Eq. X = Eq. Y by construction or of a fitted parameter being renamed as a prediction.

Assumptions & free parameters 5 free parameters · 4 assumptions · 0 invented entities

The paper's central claims are produced by fitting many-constant symbolic expressions to 65 lattice points (58 after 10% holdout) and by applying clustering and penalty hyperparameters chosen ad hoc. These fitted constants and algorithm settings, not a first-principles derivation, carry the factorization and radius conclusions. No new physical entities are introduced; the new objects are methodological.

free parameters (5)
  • Numerical constants in selected PySR expressions = e.g., 23.9, 32.5, 1.37, 0.016, 1.67, 0.638 in Eq. 38; Table I entries
    These constants are optimized by PySR against the training MSE; every physical prediction inherits them.
  • Regge exponents and coefficients in the Semi-Reggeized exemplar = 2.74, 0.087, 1.57, 0.67 in Eq. 33
    Fitted to the lattice training data; the endpoint behavior and the 'finite as x goes to 0' claim rely on these fitted numbers.
  • Expansion center c_x for ECC = 0.555
    Chosen as the midpoint of the training x-range; the Taylor coefficients, clusters, and conclusions about convergence all depend on this location.
  • Clustering hyperparameters = HDBSCAN min_samples = 42; number of clusters fixed at 3 via Kneedle
    These heuristic choices determine how many solution families are identified and which replicas are labeled clusterless.
  • Factorization penalty scale gamma = 10
    The custom loss in Eq. 28 uses gamma = 10; this scale controls how aggressively non-factorized forms are suppressed and therefore affects the FF MSE results.
assumptions (4)
  • domain assumption The first six Taylor coefficients around x = 0.555 provide a sufficient common basis to compare symbolic-regression models and to define meaningful clusters.
    Invoked in Sec. III C; the expansion point, order O((x-c_x)^5), and the use of these coefficients for clustering are choices not derived from physics.
  • domain assumption The lattice QCD matrix elements from Ref. 29 can be treated as the physical GPD H_{u-d} at xi = 0 in the fitted (x,t) region.
    Section III D describes RI/MOM renormalized quasi-GPDs; the analysis ignores the correlated systematic errors noted in Sec. IV A 4.
  • domain assumption Regge behavior x^alpha and Brodsky-Farrar counting rule (1-x)^beta are the correct endpoint forms for the isovector GPD.
    Section III E 3 builds the Semi-Reggeized ansatz on these physics rules; the claimed endpoint behavior is therefore assumed, not discovered.
  • ad hoc to paper K-means with Kneedle elbow and HDBSCAN clustering of Taylor coefficients identifies physically meaningful solution families.
    The number of clusters and the min_samples value are heuristic (Secs. III C and IV A); no independent evidence links these clusters to distinct proton structure scenarios.

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Cite this review

Pith. "Pith review of Generalized Parton Distributions from Symbolic Regression." pith.science (2026). https://pith.science/paper/NVUGLWKF

@misc{pith2026250413289,
  author       = {Pith},
  title        = {Pith review of: Generalized Parton Distributions from Symbolic Regression},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NVUGLWKF}},
  note         = {Machine review of arXiv:2504.13289}
}
abstract

AI/ML informed Symbolic Regression is the next stage of scientific modeling. We utilize a highly customizable symbolic regression package ``PySR" to model the $x$ and $t$ dependence of the flavor isovector combination $H_{u-d}(x,t,\xi)$ at $\xi=0$. These PySR models were trained on GPD results provided by both Lattice QCD and phenomenological sources GGL, GK, and VGG. We demonstrate, for the first time, the consistency and systematic convergence of Symbolic Regression by quantifying the disparate models through their Taylor expansion coefficients. In addition to PySR penalizing models with higher complexity and mean-squared error, we implement schemes that test specific physics hypotheses, including force-factorized $x$ and $t$ dependence and Regge behavior in PySR GPDs. We show that PySR can identify factorizing GPD sources based on their response to the Force-Factorized model. Knowing the precise behavior of the GPDs, and their uncertainties in a wide range in $x$ and $t$, crucially impacts our ability to concretely and quantitatively predict hadronic spatial distributions and their derived quantities.

Figures

Figures reproduced from arXiv: 2504.13289 by the authors.

Figure 1
Figure 1. FIG. 1: GPD Feynman diagram at tree level, illustrating the kinematics for a deeply virtual exclusive scattering [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Illustration of the procedure we used to investigate the properties of GPDs. To gain insight into the [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Ensemble of 1000 unbiased fits to the lattice GPD data using PySR for [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (17 more)
Figure 4
Figure 4. Figure 4: FIG. 4: K-means Within-Cluster Sum-of-Squares as a function of the number of clusters. Clustering is simultaneously [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: Cluster Color-Coded spaghetti plot of Best-Fit PySR replicas before (left) and after (right) selecting onto [PITH_FULL_IMAGE:figures/full_fig_p014_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7: Distribution of finite-moment PySR Best-Fit replicas [PITH_FULL_IMAGE:figures/full_fig_p015_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8: Ensemble of 1000 factorized fits to the LQCD GPD data using PySR for [PITH_FULL_IMAGE:figures/full_fig_p016_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9: K-means Within-Cluster Sum-of-Squares as a function of the number of clusters. The location of the elbow, [PITH_FULL_IMAGE:figures/full_fig_p016_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10: Cluster Color-Coded spaghetti plot of FF before (left) and after (right) selecting onto replicas with finite [PITH_FULL_IMAGE:figures/full_fig_p017_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11: Distribution of remaining [PITH_FULL_IMAGE:figures/full_fig_p017_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12: BF SR results for the flavor separated electromagnetic form factor, [PITH_FULL_IMAGE:figures/full_fig_p019_12.png]
Figure 13
Figure 13. Figure 13: FIG. 13: Ratios [PITH_FULL_IMAGE:figures/full_fig_p021_13.png]
Figure 14
Figure 14. Figure 14: FIG. 14: Array of fits to the lattice source data [29] for [PITH_FULL_IMAGE:figures/full_fig_p022_14.png]
Figure 15
Figure 15. Figure 15: FIG. 15: Array of fits to the VGG [59] (left) GGL [42] (center) and GK [60] (right) source data for [PITH_FULL_IMAGE:figures/full_fig_p023_15.png]
Figure 16
Figure 16. Figure 16: FIG. 16: Histogram of the MSEs relative to the testing dataset for 100 BF and FF replicas trained on Phenomenological [PITH_FULL_IMAGE:figures/full_fig_p025_16.png]
Figure 17
Figure 17. Figure 17: FIG. 17 [PITH_FULL_IMAGE:figures/full_fig_p026_17.png]
Figure 18
Figure 18. Figure 18: FIG. 18: Studying the t-dependence and corresponding [PITH_FULL_IMAGE:figures/full_fig_p027_18.png]
Figure 19
Figure 19. Figure 19: FIG. 19: The BF Cluster 1 model compared to an example FF model as a function of [PITH_FULL_IMAGE:figures/full_fig_p028_19.png]
Figure 20
Figure 20. Figure 20: FIG. 20: The average radii [PITH_FULL_IMAGE:figures/full_fig_p028_20.png]

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    Interpolation and Extrapolation We can see from Fig. 3 that not all replicas extrapolate to one of three clusters; There are spurious solutions that extrapolate in-between bands of overlapping models. To account for PySR models that do not fall into any cluster of solutions, we considered other clustering algorithms which do not require every single repli...

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    Forced Factorization For the Forced Factorization (FF) criterion, following the procedure described in Section III C, we generated 10 3 FF replicas PySR replicas and we performed an ECC (Section III B), on the distribution of replica Taylor coefficients in the PDF limit near x =cx = 0.555 to O(x−cx)5. We see from Fig. 8 that most FF replicas describe the ...

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