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arxiv: 2606.10913 · v1 · pith:GP3GOFTInew · submitted 2026-06-09 · 💻 cs.LG · stat.ML

Conservation Laws from Data Symmetry in Neural Networks

Jakob Galley , Vahid Shahverdi , Axel Flinth This is my paper

Pith reviewed 2026-06-27 13:52 UTC · model grok-4.3

classification 💻 cs.LG stat.ML
keywords neural networksconserved quantitiesdata symmetrygradient flowmean squared errorintegrals of motiontensorizable networks
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The pith

Data symmetries do not generically induce extra conserved quantities during neural network gradient-flow training.

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

The paper asks whether symmetries intrinsic to the training data create additional conserved quantities, or integrals of motion, as neural networks evolve under gradient flow. For loss functions that are analytic and non-polynomial, the authors prove that such symmetries produce no extra conservation laws in generic cases. In contrast, when the loss is mean squared error, data augmentation can create additional conserved quantities in identifiable situations. The analysis relies on a class of networks in which parameter and input effects separate through an intermediate representation.

Core claim

Under the assumption that the loss function is analytic and non-polynomial, data symmetries generically do not induce any additional integrals of motion. For mean squared error loss, there are situations in which data augmentation yields extra conserved quantities. The authors introduce tensorizable networks to characterize the architectures and losses for which this occurs.

What carries the argument

Tensorizable networks, a family of architectures in which dependence on parameters and inputs can be separated using an intermediate representation.

Load-bearing premise

The loss function is analytic and non-polynomial.

What would settle it

Finding an analytic non-polynomial loss function together with a data symmetry that produces an observable extra integral of motion during gradient flow would falsify the generic negative claim.

Figures

Figures reproduced from arXiv: 2606.10913 by Axel Flinth, Jakob Galley, Vahid Shahverdi.

Figure 1
Figure 1. Figure 1: A display of how data symmetry can give rise to conservation laws. The top row shows the ordinary and group-augmented loss landscapes for a two-parameter neural network, with the corresponding training data shown below. permutations, rotations, reflections, or other group ac￾tions. Moreover, if the learning target is known to have a symmetry a priori – meaning that it is equivariant with respect to a group… view at source ↗
Figure 2
Figure 2. Figure 2: Evolution of the deviation |I(Wt) − I(W0)| over gradient descent steps for the linear model with C3-augmentation. The true augmented dynamics preserves the direction of WΠ1⊥ . As mentioned in Section 5.1, this gives rise to the integral of motion I(W) in (35). In [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: Rotation orbit of a grid cell. with respect to the basis vℜ and vℑ. Using the same basis, we have the following H ≃    h √ 3 3 1 vℜ vℑ i   1 0 0 0 a ∓b 0 b ±a   h √ 3 3 1 vℜ vℑ i⊤ [PITH_FULL_IMAGE:figures/full_fig_p030_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The four vectors in Equation 172. Note that rotating [PITH_FULL_IMAGE:figures/full_fig_p031_5.png] view at source ↗
read the original abstract

We explore whether intrinsic symmetries of the training data lead to conserved quantities during gradient-flow training of neural networks. Under the assumption that the loss function is analytic and non-polynomial, we prove that data symmetries generically do not induce any additional integrals of motion. For mean squared error (MSE) loss, on the other hand, there are situations in which data augmentation yields extra conserved quantities. We build a framework, utilizing \emph{tensorizable networks} to describe this phenomenon. Tensorizable networks are a family of architectures whose dependence on parameters and inputs can be separated using an intermediate representation. They include linear and polynomial networks, as well as Lightning Attention.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript explores whether intrinsic symmetries of the training data induce conserved quantities during gradient-flow training of neural networks. Under the assumption that the loss function is analytic and non-polynomial, it proves that data symmetries generically do not induce any additional integrals of motion. For mean squared error (MSE) loss, however, there are situations in which data augmentation yields extra conserved quantities; these are characterized via a framework of tensorizable networks (whose parameter and input dependence can be separated via an intermediate representation), which includes linear and polynomial networks as well as Lightning Attention.

Significance. If the central claims hold, the work provides a precise distinction between loss classes regarding the emergence of conservation laws from data symmetries, together with a new architectural framework for the positive (MSE) case. The explicit conditioning on analytic non-polynomial losses and the separate treatment of MSE are strengths; the paper also supplies a mathematical proof under stated assumptions rather than empirical fitting.

minor comments (3)
  1. §2 (or wherever the tensorizable-network definition appears): the separation of parameter and input dependence via the intermediate representation should be stated as a formal definition with explicit conditions on the intermediate map, to make the scope of the framework unambiguous for readers outside the authors' immediate circle.
  2. The phrase 'generically' in the main negative result is used without a preceding sentence that recalls the precise measure or topology with respect to which genericity is taken; adding one sentence would improve readability without altering the argument.
  3. Figure captions (or the Lightning Attention example) could usefully include a one-sentence reminder of which loss is being used, to avoid any momentary confusion between the generic and MSE cases.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending minor revision. The provided summary accurately captures our central results on the distinction between analytic non-polynomial losses (where data symmetries generically yield no additional conserved quantities) and the MSE case (where tensorizable networks can produce extra integrals of motion).

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper presents a mathematical proof under the explicit assumption that the loss function is analytic and non-polynomial, showing that data symmetries generically do not induce additional integrals of motion, with a separate case for MSE loss handled via the tensorizable networks framework. No load-bearing steps reduce by construction to fitted parameters, self-definitions, or self-citation chains; the central claims are conditioned on stated assumptions and appear self-contained against external mathematical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim depends on the analytic non-polynomial assumption for the negative result and the definition of tensorizable networks for the positive MSE case.

axioms (1)
  • domain assumption The loss function is analytic and non-polynomial
    This is the key assumption for the generic result that no additional integrals of motion are induced.
invented entities (1)
  • tensorizable networks no independent evidence
    purpose: A family of architectures to describe the MSE case where extra conserved quantities appear
    Introduced in the paper to separate dependence on parameters and inputs, including linear, polynomial networks, and Lightning Attention.

pith-pipeline@v0.9.1-grok · 5633 in / 1210 out tokens · 23777 ms · 2026-06-27T13:52:17.428961+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

42 extracted references · 3 canonical work pages · 1 internal anchor

  1. [1]

    Advances in neural information processing systems , volume=

    Abide by the law and follow the flow: Conservation laws for gradient flows , author=. Advances in neural information processing systems , volume=

  2. [2]

    arXiv preprint arXiv:2405.12888 , year=

    Keep the momentum: Conservation laws beyond Euclidean gradient flows , author=. arXiv preprint arXiv:2405.12888 , year=

    arXiv

  3. [3]

    Convergence of Large Margin Separable Linear Classification , url =

    Zhang, Tong , booktitle =. Convergence of Large Margin Separable Linear Classification , url =

  4. [4]

    International Conference on Machine Learning , pages=

    Hidden symmetries of ReLU networks , author=. International Conference on Machine Learning , pages=. 2023 , organization=

    2023

  5. [5]

    arXiv preprint arXiv:1312.6120 , year=

    Exact solutions to the nonlinear dynamics of learning in deep linear neural networks , author=. arXiv preprint arXiv:1312.6120 , year=

    Pith/arXiv arXiv

  6. [6]

    Advances in neural information processing systems , volume=

    Algorithmic regularization in learning deep homogeneous models: Layers are automatically balanced , author=. Advances in neural information processing systems , volume=

  7. [7]

    arXiv preprint arXiv:1810.02032 , year=

    Gradient descent aligns the layers of deep linear networks , author=. arXiv preprint arXiv:1810.02032 , year=

    Pith/arXiv arXiv

  8. [8]

    International conference on machine learning , pages=

    Group equivariant convolutional networks , author=. International conference on machine learning , pages=. 2016 , organization=

    2016

  9. [9]

    arXiv preprint arXiv:2104.13478 , year=

    Geometric deep learning: Grids, groups, graphs, geodesics, and gauges , author=. arXiv preprint arXiv:2104.13478 , year=

    Pith/arXiv arXiv

  10. [10]

    Journal of Machine Learning Research , volume=

    A group-theoretic framework for data augmentation , author=. Journal of Machine Learning Research , volume=

  11. [11]

    Advances in neural information processing systems , volume=

    Scalars are universal: Equivariant machine learning, structured like classical physics , author=. Advances in neural information processing systems , volume=

  12. [12]

    arXiv preprint, arXiv:2005.00178 , year=

    On the benefits of invariance in neural networks , author=. arXiv preprint, arXiv:2005.00178 , year=

    arXiv 2005

  13. [13]

    Advances in Neural Information Processing Systems , volume=

    On the implicit bias of linear equivariant steerable networks , author=. Advances in Neural Information Processing Systems , volume=

  14. [14]

    Conference on Learning Theory , pages=

    Learning with invariances in random features and kernel models , author=. Conference on Learning Theory , pages=. 2021 , organization=

    2021

  15. [15]

    Transactions on Machine Learning Research , year=

    Optimization dynamics of equivariant and augmented neural networks , author=. Transactions on Machine Learning Research , year=

  16. [16]

    2025 International Conference on Sampling Theory and Applications (SampTA) , pages=

    Data augmentation and regularization for learning group equivariance , author=. 2025 International Conference on Sampling Theory and Applications (SampTA) , pages=. 2025 , organization=

    2025

  17. [17]

    2013 , publisher=

    Representation theory: a first course , author=. 2013 , publisher=

    2013

  18. [18]

    1946 , publisher=

    The classical groups: their invariants and representations , author=. 1946 , publisher=

    1946

  19. [19]

    arXiv preprint arXiv:2506.13714 , year=

    Understanding Learning Invariance in Deep Linear Networks , author=. arXiv preprint arXiv:2506.13714 , year=

    arXiv

  20. [20]

    arXiv preprint arXiv:2104.05508 , year=

    Noether: The more things change, the more stay the same , author=. arXiv preprint arXiv:2104.05508 , year=

    arXiv

  21. [21]

    Advances in Neural Information Processing Systems , volume=

    Noether’s learning dynamics: Role of symmetry breaking in neural networks , author=. Advances in Neural Information Processing Systems , volume=

  22. [22]

    arXiv preprint arXiv:2012.04728 , year=

    Neural mechanics: Symmetry and broken conservation laws in deep learning dynamics , author=. arXiv preprint arXiv:2012.04728 , year=

    arXiv 2012

  23. [23]

    arXiv preprint arXiv:2210.17216 , year=

    Symmetries, flat minima, and the conserved quantities of gradient flow , author=. arXiv preprint arXiv:2210.17216 , year=

    arXiv

  24. [24]

    International Conference on Learning Representations , year=

    A Convergence Analysis of Gradient Descent for Deep Linear Neural Networks , author=. International Conference on Learning Representations , year=

  25. [25]

    Information and Inference: A Journal of the IMA , volume=

    Learning deep linear neural networks: Riemannian gradient flows and convergence to global minimizers , author=. Information and Inference: A Journal of the IMA , volume=. 2022 , publisher=

    2022

  26. [26]

    Constructive Approximation , volume=

    Neural network identifiability for a family of sigmoidal nonlinearities , author=. Constructive Approximation , volume=. 2022 , publisher=

    2022

  27. [27]

    The Fourteenth International Conference on Learning Representations , year=

    Learning on a razor’s edge: Identifiability and singularity of polynomial neural networks , author=. The Fourteenth International Conference on Learning Representations , year=

  28. [28]

    arXiv preprint arXiv:2601.21645 , year=

    Identifiable Equivariant Networks are Layerwise Equivariant , author=. arXiv preprint arXiv:2601.21645 , year=

    Pith/arXiv arXiv

  29. [29]

    arXiv preprint arXiv:2408.17221 , year=

    Geometry of lightning self-attention: Identifiability and dimension , author=. arXiv preprint arXiv:2408.17221 , year=

    Pith/arXiv arXiv

  30. [30]

    parametric equivalence of ReLU networks , author=

    Functional vs. parametric equivalence of ReLU networks , author=. International conference on learning representations , year=

  31. [31]

    2003 , publisher=

    Linear algebra and its applications , author=. 2003 , publisher=

    2003

  32. [32]

    arXiv preprint arXiv:2501.18915 , year=

    Algebra Unveils Deep Learning--An Invitation to Neuroalgebraic Geometry , author=. arXiv preprint arXiv:2501.18915 , year=

    arXiv

  33. [33]

    arXiv preprint arXiv:2603.29566 , year=

    The Geometry of Polynomial Group Convolutional Neural Networks , author=. arXiv preprint arXiv:2603.29566 , year=

    arXiv

  34. [34]

    Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse , pages =

  35. [35]

    arXiv preprint arXiv:2605.03601 , year=

    Most ReLU Networks Admit Identifiable Parameters , author=. arXiv preprint arXiv:2605.03601 , year=

    Pith/arXiv arXiv

  36. [36]

    arXiv preprint math-ph/0609050 , year=

    How to generate random matrices from the classical compact groups , author=. arXiv preprint math-ph/0609050 , year=

    Pith/arXiv arXiv

  37. [37]

    Experimental mathematics , volume=

    Packing lines, planes, etc.: Packings in Grassmannian spaces , author=. Experimental mathematics , volume=. 1996 , publisher=

    1996

  38. [38]

    Characterizing

    Gunasekar, Suriya and Lee, Jason and Soudry, Daniel and Srebro, Nathan , month = jul, year =. Characterizing. Proceedings of the 35th

  39. [39]

    Understanding deep learning (still) requires rethinking generalization , volume =. Commun. ACM , author =. 2021 , pages =. doi:10.1145/3446776 , abstract =

    work page doi:10.1145/3446776 2021

  40. [40]

    Implicit

    Arora, Sanjeev and Cohen, Nadav and Hu, Wei and Luo, Yuping , year =. Implicit. Advances in

  41. [41]

    Neyshabur, Behnam and Tomioka, Ryota and Srebro, Nathan , month = apr, year =. In. doi:10.48550/arXiv.1412.6614 , abstract =

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.1412.6614

  42. [42]

    Soudry, Daniel and Hoffer, Elad and Nacson, Mor Shpigel and Gunasekar, Suriya and Srebro, Nathan , month = oct, year =. The. doi:10.48550/arXiv.1710.10345 , abstract =

    work page doi:10.48550/arxiv.1710.10345