Normalizing Flows to Reconstruct Pseudo-PDFs

Kostas Orginos; Yamil Cahuana Medrano

REVIEW 4 major objections 5 minor 18 references

An invertible neural network trained on Gaussian-process prior samples can reconstruct pseudo-PDFs from roughly ten Ioffe-time points while preserving physical constraints.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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T0 review · deepseek-v4-flash

2026-08-01 02:52 UTC pith:UOUYZDMW

load-bearing objection A plausible but under-validated proof of concept: the flow-based reconstruction is new, but the 'preserves physical constraints' claim needs numbers before it convinces. the 4 major comments →

arxiv 2607.25282 v1 pith:UOUYZDMW submitted 2026-07-28 hep-lat cs.LGhep-ph

Normalizing Flows to Reconstruct Pseudo-PDFs

Yamil Cahuana Medrano , Kostas Orginos This is my paper

classification hep-lat cs.LGhep-ph

keywords pseudo-PDFsIoffe-time distributionsnormalizing flowsinvertible neural networksGaussian process priorsmaximum mean discrepancylattice QCD inverse problemsparton distribution functions

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper is trying to establish that a normalizing-flow architecture, an invertible neural network, can replace the hand-tuned prior-scale step in pseudo-PDF reconstruction. Trained on functions drawn from a Gaussian-process prior, the network learns a full posterior over PDFs consistent with roughly ten Ioffe-time points. A sympathetic reader would care because lattice QCD provides only a handful of noisy Fourier-type measurements, and existing Bayesian and Gaussian-process approaches require careful prior calibration; this work asks whether that calibration can be learned instead. The paper reports closure tests showing that the network preserves unit normalization and the boundary condition q(1)=0, produces stable reconstructions, and improves as the Ioffe-time range grows. It explicitly frames the result as a proof of concept rather than a finished lattice-QCD pipeline.

Core claim

The paper's central claim is that an invertible neural network, trained as a normalizing flow on an essentially infinite dataset of PDFs sampled from a Gaussian-process prior, learns the posterior over PDFs consistent with limited Ioffe-time data. The architecture is bijective: the forward map sends a PDF to its finite-element ITD vector plus latent noise and zero padding, and the inverse map sends concatenated ITD data and latent variables back to a PDF. The authors report closure tests in which parametric PDFs are recovered stably, the normalization integral and the boundary value q(1)=0 are preserved, and reconstruction quality improves as the Ioffe-time range grows. The latent dimensions

What carries the argument

The central object is the bijective map f: R^{Nx} -> R^{Nν + Nz} built from stacked affine coupling layers. Each layer splits the input, passes one half unchanged, and applies scaling and translation networks to the other half, so the inverse can be computed analytically. Because the observable space is smaller than the PDF space, the missing dimensions are filled with latent noise variables drawn from a fixed Gaussian and zero padding; the latent dimensions carry the degrees of freedom that the data cannot determine. The training objective is a bidirectional maximum mean discrepancy loss with an inverse multi-quadratic kernel, which drives the model distribution toward the data distribution

Load-bearing premise

The load-bearing premise is that the single hand-specified Gaussian-process prior — a log-squared-exponential kernel with fixed amplitude and length scale and mean 2(1−x) — generates a training ensemble that includes the true PDF; the authors acknowledge that normalizing with respect to this prior confines the network to datasets consistent with it.

What would settle it

A concrete falsifier: hold out a PDF drawn from the same Gaussian-process prior, compute its Ioffe-time distribution, and check whether the INN's inverse-mapped ensemble contains the true PDF within its spread and satisfies normalization and q(1)=0. Failure on an in-distribution sample would overturn the central claim; failure only on out-of-prior samples would confirm the acknowledged limitation.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

If the proof of concept holds, a trained INN gives a full posterior over PDFs from roughly ten Ioffe-time points, not just a single fit, with latent sampling as the uncertainty handle.
Because the forward operator is a discretized Fourier-type transform, the same trained flow can be reused for any new ITD data consistent with the prior without retraining, only latent inference.
The paper's conclusion implies that the prior-scale hyperparameter that Gaussian-process regression must hand-tune can be absorbed into the learned flow, making reconstruction calibration part of the training.
The authors state the method extends to z-dependent pseudo-PDFs, Mellin moments, and generalized parton distributions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Inference: if the learned posterior is well calibrated, the spread across latent samples could serve as a practical uncertainty band for lattice observables, potentially cheaper than Gaussian-process regression on large grids because the expensive part is done once at training time.
Inference: the architecture is not specific to PDFs; any linear inverse problem with a truncated Fourier kernel, such as form-factor or spatial-density reconstruction, could use the same prior-flow setup provided the Gaussian-process prior covers the target class.
Inference: a testable extension is to train the prior on a mixture of kernels and means, then check recovery on validation PDFs deliberately outside the original prior; the authors' normalization argument predicts degraded extrapolation for such inputs, which would directly quantify the stated limitation.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

A plausible but under-validated proof of concept: the flow-based reconstruction is new, but the 'preserves physical constraints' claim needs numbers before it convinces.

read the letter

Short version: this is a clean proof-of-concept for using invertible flows in pseudo-PDF reconstruction, but the abstract's claim about preserving physical constraints is not backed by evidence. The architecture does not hard-code Eq. (4); it's learned through MMD against GP samples, and no residuals are reported.

What's new: as far as I can tell, this is the first time INNs are applied to the Ioffe-time reconstruction problem. The recipe—GP prior to generate a training ensemble, RealNVP affine coupling, MMD loss in both directions—is a sensible combination of established tools. The method section is clear enough that a competent grad student could reimplement it. The authors are honest about the prior-dependence limitation and the latent-dimension sensitivity; they don't oversell it as a resolved reconstruction method.

Soft spots: the stress-test note lands. Eqs. (7)–(8) are an affine coupling map with no explicit constraint enforcement. ∫q=1 and q(1)=0 are properties of the GP samples, not of the network. MMD with finite batches and an IMQ kernel is an approximation; it can leave systematic violations. The paper states 'the architecture successfully preserves the GP prior constraints' (Sec. 4) but gives no numbers. That's a load-bearing omission because the abstract promises 'preserves physical constraints.' If the constraint preservation is approximate, then the method is closer to an unconstrained regressor, and the claimed advantage over earlier Bayesian/GP approaches is not established. Second, the results are entirely qualitative—Figures 2 and 3 have no error bars, no L2 errors, no comparison to the cited GP or Bayesian baselines. For a proof of concept that's acceptable, but it limits what we can conclude. Third, the closure test uses a single parametric function (Eq. 12) and only two settings (latent dimension, ν range). It doesn't test robustness across multiple priors or PDF shapes, which matters because the training data come from the GP prior.

None of this is fatal. The paper is internally consistent, the code seems reproducible in principle, and the authors flag the main limitations themselves. The central idea—that latent variables can absorb the missing degrees of freedom and that MMD can pull samples toward the constrained space—is plausible. It just needs a more rigorous demonstration.

Who it's for: lattice-QCD people working on PDF reconstruction and ML scientists interested in inverse problems with non-injective maps. It would be a reasonable workshop/conference paper; for a journal I'd want the quantitative baseline comparison and constraint-residual analysis first.

I'd send it to peer review, but with a clear expectation that the constraint-preservation claim be tested and reported.

Referee Report

4 major / 5 minor

Summary. The paper proposes an invertible neural network (INN) trained on samples from a Gaussian-process prior conditioned to satisfy normalization and q(1)=0, in order to reconstruct PDFs from limited Ioffe-time data. The inverse mapping is trained with an MMD loss evaluated in both directions, with latent variables accounting for the missing information of the inverse problem. A closure test compares the INN reconstruction of a parametric beta-like PDF at different latent dimensions and different ranges of Ioffe time. The paper claims that the architecture preserves physical constraints and extrapolation properties, and that latent variables encode the missing degrees of freedom.

Significance. If substantiated, the approach would be a useful learned alternative to existing GP/Bayesian reconstruction methods, with the latent variables explicitly representing the ill-posed directions of the inverse problem. The authors are to be credited for testing on an independently defined parametric closure target rather than on training samples, and for clearly acknowledging the prior-coverage restriction. The GP-prior construction and the architectural details are transparent. However, at present the offered evidence is qualitative: the central claims of constraint preservation, extrapolation behavior, and equivalence to previous approaches are not backed by quantitative metrics or comparisons. No code or data is made available, which limits reproducibility of the figures.

major comments (4)

[§3, Figs. 2–3; §2, Eqs. (7)–(11)] The main evidence is visual only. There are no error bars, no residuals for the constraints in Eq. (4), and no accuracy metric between qINN and qval. This matters because the architecture does not enforce normalization or q(1)=0 structurally: Eq. (7)–(8) is a standard affine coupling map, and the only mechanism pulling outputs toward the constraints is the MMD loss in Eq. (11) against GP-prior samples. Finite-batch MMD with an IMQ kernel matches distributions only approximately. The paper should report, for the closure test, the mean and spread of ∫qINN − 1 and qINN(x=1), plus at least an L2 or MMD error versus qval, as a function of Nnoise and the available ν range. Without these numbers, the advertised advantage over unconstrained regression is not established.
[§4 Discussion; §5 Conclusion] The text repeatedly states that the INN leads to 'similar conclusions' to Gaussian-process regression and earlier Bayesian methods, but no comparison to the cited baselines (Dutrieux et al. 2025, Karpie et al. 2019, Medrano et al. 2026) is made on the same closure test. A quantitative benchmark using the same ITD grid and the same qval would be needed to support the claim that this is an alternative rather than merely a different architecture. Without such a comparison, the novelty and practical value of the method remain a conjecture.
[§2, Eq. (5); §4 Discussion] The training distribution is a single GP prior with fixed mean 2(1−x), amplitude 30, and log-RBF length scale 0.72. As the authors note, the INN is then only reliable for data consistent with this prior. This is not just a caveat but a load-bearing premise: the closure PDF in Eq. (12) is a beta-like curve with α+1 ≈ 0.8 and β+1 ≈ 4, whose mean is far from the prior mean. The paper gives no prior-predictive check or sensitivity analysis showing that the training ensemble actually covers qval. I recommend adding a diagnostic (e.g., the prior density of qval under Eq. (5), or posterior predictive ITD residuals for qval) and a test with a deliberately out-of-prior target to define the failure mode.
[§3, Fig. 2] The sensitivity to Nnoise is shown but not characterized. The text states that the reconstruction depends on the latent dimension in the low-x/high-ν region, yet no quantitative analysis of this dependence or a criterion for choosing Nnoise is provided. This is a central hyperparameter of the method, and the paper should report reconstruction error versus Nnoise, rather than only showing that curves differ visually.

minor comments (5)

[§2, Eq. (5)] The Lagrange-multiplier conditioning formula is typeset ambiguously; K and μ should be written with explicit inverses and dimensions so that the conditioning step is unambiguous.
[§3, Eq. (12) and Figs. 2–3] It is unclear whether α and β are fixed values or random draws in the displayed closure test. If random, a single draw is not enough to assess variability; if fixed, the values should be stated explicitly.
[Figures] Figures 2 and 3 would benefit from axis labels, legends, and error bands or uncertainty quantification; the current captions do not describe what curves represent beyond 'different reconstructions'.
[§2] Please report the number of training samples, MMD batch size, number of independent training runs, and the random seed policy used for the displayed results; this is necessary for reproducibility.
[References] The companion work by Medrano et al. 2026 is cited; if it is the GP baseline to which this paper should be compared, this should be made explicit in Section 4.

Circularity Check

0 steps flagged

No circular reduction: the closure test is independent of the training ensemble; only a minor self-citation supplies the prior scale.

full rationale

The central claim is tested by a genuine closure test. The INN is trained on forward-simulated samples drawn from the constrained GP prior (Eqs. 3–5), and the closure PDF q_val in Eq. (12) is generated independently and is not used to fit network weights, select GP hyperparameters, or choose the noise dimension. Using the same forward operator B in training and testing is the standard structure of an inverse problem, not a circular reuse of the target. The statement that the architecture 'preserves physical constraints' is weaker than claimed: the INN is unconstrained and constraint satisfaction is only induced by the MMD loss (Eq. 11) against constrained prior samples, with no numerical residuals reported. That is an evidentiary gap, not a circular reduction, and the paper itself partially acknowledges the prior-dependence limitation in Section 4. The prior scale is imported from the authors' own earlier work (Dutrieux et al. 2025; Medrano et al. 2026), but this is a modeling input and does not force the closure result; no prediction is defined in terms of the target, and no fitted parameter is renamed as a prediction.

Axiom & Free-Parameter Ledger

6 free parameters · 6 axioms · 0 invented entities

The central reconstruction depends on several hand-chosen quantities: the GP kernel amplitude (30), length scale (0.72), mean (2(1-x)), the latent noise dimension, the loss weights, and the MMD kernel scale. These are not fitted to the closure data but are prior/architecture choices. The key domain assumptions are that Eq. (2) captures the physics and that the GP prior is representative.

free parameters (6)

GP kernel amplitude = 30
Amplitude of log-RBF kernel in Eq. (5); sets the variance of the prior PDF ensemble; chosen by hand (following Dutrieux et al. 2025), not fitted to the closure data.
GP log-RBF length scale = 0.72
Denominator in exp(-|log x - log x'|^2 / 0.72); controls smoothness; hand-selected.
GP prior mean mu0(x) = 2(1-x)
Mean of the Gaussian process in Eq. (5); encodes a valence-like shape; hand-chosen.
Latent/noise dimension Nnoise = not stated (varied in Fig. 2)
The number of noise variables in the INN; the paper shows reconstructions depend on it and says an optimal configuration needs further analysis (Section 4).
MMD kernel active scale sigma = {0.25s, 0.5s, s, 2s, 4s} with s=median pairwise distance
IMQ kernel scale in Eq. (10); the active parameter is selected from this set; the procedure is not fully specified.
Loss weights = lambda_x=lambda_y=lambda_z=1/3, lambda_pad=1/6
Weights in Eq. (11) chosen by hand; no sensitivity analysis.

axioms (6)

domain assumption The pseudo-PDF inverse problem reduces to Eqs. (2): Re M and Im M are cosine/sine transforms of q^+ and q^-; z-dependence is ignored.
Used throughout; the paper states 'we will not consider the z dependence of PDF, because the matching procedure does not interfere with the inverse problem at hand' (Introduction).
domain assumption Physical PDFs satisfy normalization integral = 1 and boundary q(1)=0 (Eq. 4).
Imposed on the GP prior via Lagrange multipliers; the reconstruction inherits these constraints from training data.
ad hoc to paper The log-RBF GP prior with amplitude 30, length scale 0.72 and mean 2(1-x) generates a representative ensemble of physical PDFs.
The whole INN is trained on samples from this prior; if real PDFs lie outside this set, the posterior is biased (the authors acknowledge this in Section 4).
ad hoc to paper Six affine coupling layers with 3-layer MLPs (hidden 256, LeakyReLU(0.2)) are expressive enough to represent the posterior.
No expressive-capacity or convergence analysis is provided; the architecture is taken from RealNVP/Ardizzone.
standard math MMD with the IMQ kernel (Eq. 10) is a valid and sufficient training objective for posterior approximation.
MMD is a standard distance on probability measures; the paper cites Mehraban and Pichler 2025.
standard math Second-order finite elements accurately approximate the Fourier integrals in Eq. (2).
The discrete forward operator B is built this way; no convergence study is reported.

pith-pipeline@v1.3.0-alltime-deepseek · 5374 in / 14060 out tokens · 122550 ms · 2026-08-01T02:52:09.104197+00:00 · methodology

0 comments

read the original abstract

We investigate a normalizing-flow approach for reconstructing parton distribution functions (PDFs) from synthetic matrix-element data. Our framework combines Gaussian Process priors with invertible neural networks to learn a posterior distribution over PDFs consistent with limited Ioffe-time data. We demonstrate that the architecture preserves physical constraints and extrapolation properties.

Figures

Figures reproduced from arXiv: 2607.25282 by Kostas Orginos, Yamil Cahuana Medrano.

**Figure 2.** Figure 2: Comparison of different reconstructions of the PDF varying the noise dimensions. [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Figure 3: Comparison of different reconstructions of the PDF varying the ITD data available. [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

discussion (0)

Reference graph

Works this paper leans on

18 extracted references · 5 canonical work pages

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George Papamakarios, Eric Nalisnick, Danilo Jimenez Rezende, Shakir Mohamed, and Balaji Lakshminarayanan. Normalizing flows for probabilistic modeling and inference, 2021. URL https://arxiv.org/abs/1912.02762

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doi:10.1103/physrevlett.110.262002 2013

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work page doi:10.1007/jhep04(2019)057 2019

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Luigi Del Debbio, Tommaso Giani, Joseph Karpie, Kostas Orginos, Anatoly Radyushkin, and Savvas Zafeiropoulos. Neural-network analysis of parton distribution functions from ioffe-time pseudodistributions. Journal of High Energy Physics, 2021 0 (2), February 2021. ISSN 1029-8479. doi:10.1007/jhep02(2021)138. URL http://dx.doi.org/10.1007/JHEP02(2021)138

work page doi:10.1007/jhep02(2021)138 2021

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Parton distributions from lattice data: the nonsinglet case

Krzysztof Cichy, Luigi Del Debbio, and Tommaso Giani. Parton distributions from lattice data: the nonsinglet case. Journal of High Energy Physics, 2019 0 (10), October 2019. ISSN 1029-8479. doi:10.1007/jhep10(2019)137. URL http://dx.doi.org/10.1007/JHEP10(2019)137

work page doi:10.1007/jhep10(2019)137 2019

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Parton distribution functions from lattice qcd using bayes-gauss-fourier transforms

Constantia Alexandrou, Giovanni Iannelli, Karl Jansen, and Floriano Manigrasso. Parton distribution functions from lattice qcd using bayes-gauss-fourier transforms. Physical Review D, 102 0 (9), November 2020. ISSN 2470-0029. doi:10.1103/physrevd.102.094508. URL http://dx.doi.org/10.1103/PhysRevD.102.094508

work page doi:10.1103/physrevd.102.094508 2020

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Bayesian inference with gaussian processes for the determination of parton distribution functions, 2024

Alessandro Candido, Luigi Del Debbio, Tommaso Giani, and Giacomo Petrillo. Bayesian inference with gaussian processes for the determination of parton distribution functions, 2024. URL https://arxiv.org/abs/2404.07573

Pith/arXiv arXiv 2024

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Gaussian processes for inferring parton distributions, 2026

Yamil Cahuana Medrano, Hervé Dutrieux, Joseph Karpie, Kostas Orginos, and Savvas Zafeiropoulos. Gaussian processes for inferring parton distributions, 2026. URL https://arxiv.org/abs/2510.21041

arXiv 2026

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Simple nonparametric reconstruction of parton distributions from limited Fourier information

Herv \'e Dutrieux, Joseph Karpie, Kostas Orginos, and Savvas Zafeiropoulos. Simple nonparametric reconstruction of parton distributions from limited Fourier information . Phys. Rev. D, 111 0 (3): 0 034515, 2025. doi:10.1103/PhysRevD.111.034515

work page doi:10.1103/physrevd.111.034515 2025

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Pellegrini, Ralf S

Lynton Ardizzone, Jakob Kruse, Sebastian Wirkert, Daniel Rahner, Eric W. Pellegrini, Ralf S. Klessen, Lena Maier-Hein, Carsten Rother, and Ullrich Köthe. Analyzing inverse problems with invertible neural networks, 2019. URL https://arxiv.org/abs/1808.04730

Pith/arXiv arXiv 2019

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Normalizing flows for probabilistic modeling and inference, 2021

George Papamakarios, Eric Nalisnick, Danilo Jimenez Rezende, Shakir Mohamed, and Balaji Lakshminarayanan. Normalizing flows for probabilistic modeling and inference, 2021. URL https://arxiv.org/abs/1912.02762

Pith/arXiv arXiv 2021

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Density estimation using real nvp, 2017

Laurent Dinh, Jascha Sohl-Dickstein, and Samy Bengio. Density estimation using real nvp, 2017. URL https://arxiv.org/abs/1605.08803

Pith/arXiv arXiv 2017

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Deep Learning - Foundations and Concepts

Christopher Michael Bishop and Hugh Bishop. Deep Learning - Foundations and Concepts. 1 edition, 2023. ISBN 978-3-031-45468-4. doi:https://doi.org/10.1007/978-3-031-45468-4

doi:10.1007/978-3-031-45468-4 2023

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Quantization of probability measures in maximum mean discrepancy distance, 2025

Zahra Mehraban and Alois Pichler. Quantization of probability measures in maximum mean discrepancy distance, 2025. URL https://arxiv.org/abs/2503.11868

Pith/arXiv arXiv 2025

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Kingma and Jimmy Ba

Diederik P. Kingma and Jimmy Ba. Adam: A method for stochastic optimization, 2017. URL https://arxiv.org/abs/1412.6980

Pith/arXiv arXiv 2017

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Sgdr: Stochastic gradient descent with warm restarts, 2017

Ilya Loshchilov and Frank Hutter. Sgdr: Stochastic gradient descent with warm restarts, 2017. URL https://arxiv.org/abs/1608.03983

Pith/arXiv arXiv 2017