Pith. sign in

REVIEW 3 major objections 4 minor 1 cited by

Learning Broken Symmetries with Approximate Invariance

T0 review · 3 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read A hybrid network learns broken symmetries as quickly as exactly invariant networks while avoiding their performance ceiling.

desk verdict A clean, honest toy demonstration that a learned pT-dependent hybrid of invariant and general subnetworks gets the best of both worlds, but the Conclusion overclaims a pT-free variant that Appendix C never actually tests. read the letter →

arxiv 2412.18773 v2 pith:V6O2OPCV submitted 2024-12-25 hep-ph cs.LGhep-ex

Seth Nabat , Aishik Ghosh , Edmund Witkowski , Gregor Kasieczka , Daniel Whiteson This is my paper · ORCID
classification hep-phcs.LGhep-ex
keywords approximatesymmetrylearnedfactorhybriddual-subnetarchitectureLorentzinvarianceparticleflownetworkZbosonclassificationtransversemomentumresolutionbroken
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 argues that when a physical symmetry is only approximately respected by data, neither an exactly invariant network nor a fully unconstrained network is the right tool: the former learns fast but cannot represent the broken symmetry, the latter can represent it but learns slowly. It proposes a hybrid network made of an invariant subnet and a symmetry-agnostic subnet, whose outputs are blended by a learned symmetry factor that depends on the transverse momentum. In a simplified Z-boson-to-muon classification task where detector resolution smears the reconstructed mass more at high momentum, the hybrid learns as quickly as the invariant network while reaching the asymptotic performance of the unconstrained network. If the result generalizes, it gives a practical way to build approximate equivariance into neural networks for data with imperfect symmetries.

What carries the argument

The load-bearing object is the hybrid dual-subnet architecture with a learned symmetry factor. The symmetry-agnostic subnet is a particle-flow network (PFN) that consumes the full muon four-vectors; the invariant subnet is a fully connected MLP fed only with Lorentz-invariant quantities, namely the squared Minkowski norm and the Minkowski inner product. A third small MLP takes the transverse momentum and emits a softmax weight that interpolates between the two latent representations via weighted average pooling before a final classifier. The learned weight is what turns the hard constraint of full Lorentz invariance into a soft, data-dependent one: at low $p_T$, where the detector respects Lorentz symmetry, the model leans on the invariant subnet, and at high $p_T$, where resolution degrades, it leans on the general subnet.

What would settle it

Generate a toy $Z\to\mu\mu$ dataset in which the mass-resolution width is a known function of muon pseudorapidity (or of a two-momentum combination) rather than transverse momentum, and compare the explicitly $p_T$-weighted hybrid with the concatenation variant. If the $p_T$-weighted version no longer beats the exactly invariant network while the concatenation version still does, the choice of conditioning variable is the load-bearing design decision.

Watch

Extended reading notes

Core claim

The central claim is that encoding an approximate symmetry directly into a network structure, rather than enforcing it exactly or ignoring it, yields both fast learning and high asymptotic accuracy. The paper demonstrates this with a classifier for $pp\to Z\to\mu^+\mu^-$ against background, where the underlying Lorentz symmetry is broken by a momentum-dependent detector resolution. The hybrid model combines a particle-flow network that sees full muon four-vectors with a Lorentz-invariant MLP that sees only the squared Minkowski norm and the Minkowski inner product, averaging their latent representations with a weight produced by a small MLP that reads the transverse momentum. The learned weight shifts toward the general subnet as $p_T$ grows, showing the network discovers the breaking pattern. In the small-data regime the hybrid matches the invariant network's convergence speed and beats its plateau; with equal subnet capacity and large data it matches the general network's asymptotic AUC.

Load-bearing premise

The approach assumes that the way the detector breaks Lorentz symmetry is fully captured by one number, the transverse momentum, and that a small network can learn the right blend from that number.

Editorial extensions

If this is right

  • When training data are limited, the hybrid should outperform an exactly invariant network on tasks with broken symmetries, because it can represent the breaking while retaining the invariant network's fast convergence.
  • Given enough data and equal capacity for the general subnet, the hybrid's asymptotic performance converges to that of a fully unconstrained network, so the symmetry prior does not permanently cap accuracy.
  • The learned symmetry factor provides an interpretable map of where and how strongly a symmetry is broken as a function of the conditioning variable.
  • The same architecture can be applied to image tasks where pixelization or edge effects break translational invariance, and to detector data with momentum-dependent response.
  • An alternative hybrid that concatenates the subnet outputs instead of weighted-averaging them achieves comparable performance without needing the explicit $p_T$ input, at the cost of interpretability.

Reading between the lines

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

  • If the learned-weight idea transfers, it suggests that any equivariant layer could be softened by a learnable gate conditioned on the variables that break the symmetry, turning group constraints into a continuum between equivariance and full freedom.
  • The paper's success with a hand-chosen conditioning variable ($p_T$) raises the testable question of whether the MLP can learn the breaking pattern from the data itself; the concatenation variant in Appendix C is a step in that direction but was not tested on a case with unknown breaking dependence.
  • A natural stress test would be a classification task where the symmetry-breaking depends on a two-dimensional combination of momenta, which would separate the explicit-$p_T$ architecture from the concatenation variant.
  • Because the weighting is trained end-to-end, the method also provides a way to estimate the effective scale of symmetry breaking in a dataset: the learned weight as a function of the conditioning variable is a direct readout.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper proposes a dual-subnet architecture for classification in settings where an exact symmetry is only approximate. One subnet is a Lorentz-agnostic particle flow network; the other is a fully connected network built on Lorentz-invariant inputs. The two latent representations are combined by a learned weight that is a function of the muon transverse momentum. In a toy Z→μ+μ− versus background classification task with pT-dependent Gaussian smearing, the authors compare the hybrid to a general PFN and an exactly invariant network, using AUC versus training epoch on small and large datasets. They find that the hybrid matches the invariant network's rapid early learning while reaching higher asymptotic AUC, and that capacity-matched hybrids match the general network asymptotically. Appendix C explores alternative pooling strategies.

Significance. The central reported finding is a useful step toward exploiting approximate symmetries in high-energy physics and beyond. The experiments use five ensembles with standard-deviation bands, parameter-matched baselines, open data/code, and honest reporting of a divergent trial in which the learned pT-weighting had a qualitatively different shape but comparable performance. These strengths make the toy result credible. The main weakness is that the manuscript's broader generality claim—especially in the Conclusion—exceeds the tested configurations: no experiment removes the pT input from the pooling mechanism, so the rapid-learning benefit may be tied to the oracle variable rather than to the dual-subnet structure itself.

major comments (3)
  1. [Section V (Conclusions) and Appendix C] The Conclusion states that a hybrid network 'can also be designed without explicitly embedding the pT dependence' by concatenating the outputs of the two subnets, citing Appendix C. However, every concatenation variant reported there still injects pT or the pT-derived learned weight: the first variant concatenates 'post-weighted latent space representations'; the second concatenates 'the unweighted latent space dimensions and adding the learned weight as an additional input'; and the best-performing variant uses 'pT as the additional parameter instead of the learned weight.' No configuration tests pure concatenation of unweighted subnet outputs with neither pT nor a learned weight. The pT-free claim in the Conclusion is therefore not supported by the reported data. Add such a control experiment, or revise the Conclusion to claim only that concatenation is a viable alternative pooling strategy.
  2. [Section IV, paragraph beginning 'Note that, in one out of the five trials'] The observation that one of five learned-weight functions converged to a qualitatively different shape that prefers the invariant subnet at high pT, yet performed comparably, suggests that the specific pT-dependent weighting shape is not essential to the hybrid's advantage. Because the weight MLP receives pT, which is the exact variable controlling the smearing (σ = 0.1 pT, Section II), the current experiments cannot separate the contribution of the dual-subnet inductive bias from the contribution of the oracle pT input. A minimal control—for example, shuffling pT before feeding it to the learned-weight MLP, or removing pT from the weight MLP—would clarify the mechanism and is needed to support the generality of the method.
  3. [Sections II–III] The toy demonstration embeds the symmetry-breaking mechanism as a known, low-dimensional input: the Gaussian smearing width depends only on pT, and the learned-weight MLP is supplied pT. The authors acknowledge this choice ('We chose to explicitly encode the relationship between symmetry breaking and pT...'), but the paper does not test what happens when the breaking depends on variables not supplied in this way. A concrete test would be to make the smearing width depend on a nonlinear function of multiple muon-level features and to compare hybrids trained with and without access to pT; without such a test, the claimed generality to 'more complex dependence of symmetry factor on the data itself' (Section V) remains a conjecture rather than a demonstrated property.
minor comments (4)
  1. [Section II] The reweighting procedure is described as using 100 bins, while Figure 1 uses 500 bins; please state explicitly which binning was used for the training data and whether the results are sensitive to this choice.
  2. [Section III and Appendix B] The pooling equation uses a single scalar weight w, whereas Appendix B describes a learned-weight MLP with a two-dimensional output followed by a Softmax; reconcile these notations by defining w as one component of the softmax output.
  3. [Section IV] The phrase 'surprisingly preferred' editorializes an empirical result; consider replacing it with a neutral description.
  4. [Abstract and Section V] The phrase 'escapes its performance limitations' slightly overstates the large-dataset comparison, where the general network slightly outperforms the hybrid after long training; a more precise phrasing would be 'escapes the performance ceiling of the invariant network.'

Circularity Check

0 steps flagged · score 1.0 of 10

No circular derivation: the hybrid model's training and AUC comparisons are empirical, with only contextual, non-load-bearing self-citations.

full rationale

The central claim—that the hybrid dual-subnet model with a learned pT-dependent weight learns as rapidly as an invariant network while escaping its asymptotic ceiling—is supported by measured learning curves and AUC values (Figs. 4, 6, 7), not by a fitted parameter renamed as a prediction. The learned weight w is trained end-to-end by backpropagation and is later interpreted in Fig. 5; it is not used to predict a quantity it was fit to. The explicit use of pT in the learned-weight MLP mirrors the known smearing model sigma = 0.1 pT in Sec. II, which is an informed modeling choice rather than a circular embedding of the result. The only self-citations (e.g., [23], and to a lesser extent [17] with partial author overlap) are contextual background or related work and are not load-bearing for the paper's conclusions. No uniqueness theorem, no ansatz, and no prior result of the same authors is invoked to forbid alternative architectures or to force the result. Two non-circular weaknesses are worth flagging: (1) the Conclusion's statement that a hybrid 'can also be designed without explicitly embedding the pT dependence' is not established by Appendix C, since each concatenation variant either includes the pT-derived learned weight or feeds pT itself as an additional input, and (2) one of five trials produced a qualitatively different learned pT-weighting with comparable AUC, weakening the interpretation that the specific learned weighting drives the performance gain. These are generalization and interpretability limitations, not circular reasoning. No step in the derivation reduces to its inputs by construction, so no circular step is reported.

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

The central claim is an empirical demonstration on a synthetic dataset; the paper contributes a new architecture, not a new physical law. The main assumptions are the toy detector model and the availability of pT as a symmetry-breaking handle. No new particles, forces, or conserved quantities are introduced.

free parameters (3)
  • detector smearing sigma scale = 0.1 x pT_true
    Chosen by hand in Section II to model pT-dependent muon momentum resolution and exaggerate the symmetry-breaking effect for demonstration.
  • background reweighting bin count = 100 bins (500 in Fig. 1)
    Histogram binning choice in Section II to match signal and background muon pT distributions.
  • total hidden neurons per model = 1152
    Fairness constraint in Section IV equalizing model capacity across architectures; a deliberate experimental choice.
assumptions (4)
  • ad hoc to paper Detector response is modeled as Gaussian smearing of muon pT with sigma proportional to pT.
    Section II; this toy model introduces the symmetry breaking and is not validated against a real detector.
  • domain assumption Background events can be reweighted in muon pT to match signal.
    Section II; assumes no other discriminating features between signal and background besides the invariant mass spectrum.
  • domain assumption The set of Lorentz invariants (squared Minkowski norm and inner product) is sufficient for the invariant subnet.
    Section III; a standard choice for two-body decays, but an assumption about sufficiency of these inputs.
  • domain assumption The learned-weight MLP can effectively learn the pT-dependent mixing.
    Section III and Appendix C; the architecture assumes a small MLP on pT can represent the necessary weighting.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Learning Broken Symmetries with Approximate Invariance." pith.science (2026). https://pith.science/paper/V6O2OPCV

@misc{pith2026241218773,
  author       = {Pith},
  title        = {Pith review of: Learning Broken Symmetries with Approximate Invariance},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/V6O2OPCV}},
  note         = {Machine review of arXiv:2412.18773}
}
read the original abstract

Recognizing symmetries in data allows for significant boosts in neural network training, which is especially important where training data are limited. In many cases, however, the exact underlying symmetry is present only in an idealized dataset, and is broken in actual data, due to asymmetries in the detector, or varying response resolution as a function of particle momentum. Standard approaches, such as data augmentation or equivariant networks fail to represent the nature of the full, broken symmetry, effectively overconstraining the response of the neural network. We propose a learning model which balances the generality and asymptotic performance of unconstrained networks with the rapid learning of constrained networks. This is achieved through a dual-subnet structure, where one network is constrained by the symmetry and the other is not, along with a learned symmetry factor. In a simplified toy example that demonstrates violation of Lorentz invariance, our model learns as rapidly as symmetry-constrained networks but escapes its performance limitations.

Figures

Figures reproduced from arXiv: 2412.18773 by the authors.

Figure 1
Figure 1. FIG. 1. Demonstration of background reweighting to match [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Top: mass distribution for signal in three [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Model used to learn approximate Lorentz-invariant [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (5 more)
Figure 6
Figure 6. Figure 6: FIG. 6. Performance versus fraction of dataset used in training [PITH_FULL_IMAGE:figures/full_fig_p005_6.png]
Figure 4
Figure 4. Figure 4: FIG. 4. Network performance, measured by AUC, versus [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Proportion of the general network used in the hybrid [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Network performance on the large dataset, measured by [PITH_FULL_IMAGE:figures/full_fig_p006_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Hybrid network performance on the small dataset [PITH_FULL_IMAGE:figures/full_fig_p007_8.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Explicit or Implicit? Encoding Physics at the Precision Frontier

    hep-ph 2026-03 conditional novelty 6.0 of 10

    On three precision classification tasks — reweighting-based unfolding, likelihood-ratio estimation, and weakly supervised anomaly detection — a Lorentz-equivariant transformer and a pretrained foundation model perform...

Reference graph

Works this paper leans on

35 extracted references · 20 canonical work pages · cited by 1 Pith paper

  1. [1]

    Baldi, P

    P . Baldi, P . Sadowski, and D. Whiteson, Nat. Commun. 5, 4308 (2014)

    work page 2014

  2. [2]

    de Oliveira, M

    L. de Oliveira, M. Kagan, L. Mackey, B. Nachman, and A. Schwartzman, J. High Energy Phys. 07 (2016) 069

    work page 2016

  3. [3]

    Feickert and B

    M. Feickert and B. Nachman, arXiv:2102.02770

    arXiv

  4. [4]

    Alzubaidi, J

    L. Alzubaidi, J. Zhang, A. Humaidi et al., J. Big Data 8,5 3 (2021)

    work page 2021

  5. [5]

    Cohen and M

    T. Cohen and M. Welling, in Proceedings of the 33rd International Conference on Machine Learning , edited by M. F. Balcan and K. Q. Weinberger, Proceedings of Machine Learning Research V ol. 48 (PMLR, New Y ork, USA, 2016), pp. 2990 –2999, https://proceedings.mlr.press/ v48/cohenc16.html

    work page 2016

  6. [6]

    Agrawal, J

    P . Agrawal, J. Carreira, and J. Malik, in 2015 IEEE International Conference on Computer Vision (ICCV) (IEEE Computer Society, Los Alamitos, CA, USA, 2015), pp. 37 –45, ISSN 2380-7504, https://doi.ieeecom putersociety.org/10.1109/ICCV .2015.13

    doi:10.1109/iccv 2015

  7. [7]

    Gens and P

    R. Gens and P . M. Domingos, in Advances in Neural Information Processing Systems , edited by Z. Ghahramani, M. Welling, C. Cortes, N. Lawrence, and K. Weinberger (Curran Associates, Inc., 2014), V ol. 27, https://proceedings.neurips.cc/ paper_files/paper/2014/file/ f9be311e65d81a9ad8150a60844bb94c-Paper.pdf

    work page 2014

  8. [8]

    Bogatskiy, T

    A. Bogatskiy, T. Hoffman, D. W. Miller, and J. T. Offermann, arXiv:2211.00454

    arXiv

Show all 35 references
  1. [9]

    Shimmin, Z

    C. Shimmin, Z. Li, and E. Smith, arXiv:2303.11288

    arXiv

  2. [10]

    Shimmin, arXiv:2107.02908

    C. Shimmin, arXiv:2107.02908

    arXiv

  3. [11]

    Bogatskiy et al., in Snowmass 2021 (2022), arXiv:2203 .06153

    A. Bogatskiy et al., in Snowmass 2021 (2022), arXiv:2203 .06153

    2022

  4. [12]

    Köhler, L

    J. Köhler, L. Klein, and F. No´ e, arXiv:2006.02425

    2006 arXiv

  5. [13]

    F. B. Fuchs, D. E. Worrall, V . Fischer, and M. Welling, arXiv:2006.10503

    2006 arXiv

  6. [14]

    Thomas, T

    N. Thomas, T. E. Smidt, S. Kearnes, L. Yang, L. Li, K. Kohlhoff, and P . Riley, arXiv:1802.08219

    arXiv

  7. [15]

    Finzi, S

    M. Finzi, S. Stanton, P . Izmailov, and A. G. Wilson, arXiv:2002.12880

    2002 arXiv

  8. [16]

    V . G. Satorras, E. Hoogeboom, and M. Welling, arXiv:2102 .09844

  9. [17]

    Butter, G

    A. Butter, G. Kasieczka, T. Plehn, and M. Russell, SciPost Phys. 5, 028 (2018)

    2018

  10. [18]

    S. Gong, Q. Meng, J. Zhang, H. Qu, C. Li, S. Qian, W . Du, Z.-M. Ma, and T.-Y . Liu,J. High Energy Phys. 07 (2022) 030

    2022

  11. [19]

    C. Li, H. Qu, S. Qian, Q. Meng, S. Gong, J. Zhang, T.-Y . Liu, and Q. Li, Phys. Rev. D 109, 056003 (2024)

    2024

  12. [20]

    H. Bahl, N. Elmer, L. Favaro, M. Haußmann, T. Plehn, and R. Winterhalder, arXiv:2412.12069

  13. [21]

    Perez and J

    L. Perez and J. Wang, arXiv:1712.04621. FIG. 8. Hybrid network performance on the small dataset (20,000 events), measured by AUC, versus training epoch for different pooling methods. Colored bands indicate the degree of statistical variation and correspond to one standard devi...

    2025 arXiv

  14. [22]

    R. Liu, J. Lehman, P . Molino, F. P . Such, E. Frank, A. Sergeev, and J. Y osinski, arXiv:1807.03247

    arXiv

  15. [23]

    Witkowski and D

    E. Witkowski and D. Whiteson, arXiv:2311.05952

    arXiv

  16. [24]

    Alwall, R

    J. Alwall, R. Frederix, S. Frixione, V . Hirschi, F. Maltoni, O. Mattelaer, H. S. Shao, T. Stelzer, P . Torrielli, and M. Zaro, J. High Energy Phys. 07 (2014) 079

    2014

  17. [25]

    Aad et al

    G. Aad et al. (A TLAS Collaboration), Phys. Rev. D 105, 092002 (2022)

    2022

  18. [26]

    Aad et al

    G. Aad et al. (A TLAS Collaboration), arXiv:2412.01600

    arXiv

  19. [27]

    d ’Ascoli, H

    S. d ’Ascoli, H. Touvron, M. L. Leavitt, A. S. Morcos, G. Biroli, and L. Sagun, J. Stat. Mech. (2022) 114005

    2022

  20. [28]

    Finzi, G

    M. Finzi, G. W. Benton, and A. G. Wilson, arXiv:2112 .01388

  21. [29]

    Bogatskiy, B

    A. Bogatskiy, B. Anderson, J. Offermann, M. Roussi, D. Miller, and R. Kondor, in Proceedings of the 37th International Conference on Machine Learning ,e d i t e db y H. Daum´e III and A. Singh, Proceedings of Machine Learning Research V ol. 119 (PMLR, 2020), pp. 992 –1002, htt...

    2020

  22. [30]

    Brehmer, V

    J. Brehmer, V . Bresó, P . de Haan, T. Plehn, H. Qu, J. Spinner, and J. Thaler, arXiv:2411.00446

  23. [31]

    Murnane, S

    D. Murnane, S. Thais, and J. Wong, J. Phys. Conf. Ser. 2438, 012121 (2023)

    2023

  24. [32]

    P . T. Komiske, E. M. Metodiev, and J. Thaler, J. High Energy Phys. 01 (2019) 121

    2019

  25. [33]

    Loshchilov and F

    I. Loshchilov and F. Hutter, arXiv:1711.05101

    arXiv

  26. [34]

    Paszke, S

    A. Paszke, S. Gross, F. Massa, A. Lerer, J. Bradbury, G. Chanan, T. Killeen, Z. Lin, N. Gimelshein, L. Antiga et al., arXiv:1912.01703

    1912 arXiv

  27. [35]

    https://github.com/atomicsorcerer/learning-broken-symmetries SETH NABA T et al. PHYS. REV . D 111, 072002 (2025) 072002-8

    2025

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.