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Conservation Laws from Data Symmetry in Neural Networks

Jakob Galley, Vahid Shahverdi, Axel Flinth

Jun 9, 2026arXiv:2606.10913v1
cs.LGstat.ML
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#2418 of 5669 · cs.LG
Tournament Score
1419±43
10501750
56%
Win Rate
9
Wins
7
Losses
16
Matches
Rating
6.8/ 10
Significance7
Rigor8.5
Novelty7.5
Clarity7.5

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.

AI Impact Assessments

(1 models)

Scientific Impact Assessment

Core Contribution

This paper addresses a fundamental question at the intersection of geometric deep learning, optimization theory, and classical mechanics: Can symmetries in training data create new conserved quantities during gradient-flow training? The answer is nuanced and depends critically on the loss function and architecture.

The paper delivers two main theoretical results:

1. Negative result (Theorem 3): For analytic, non-polynomial margin losses (e.g., logistic, exponential) and finite symmetry groups, data symmetries generically do *not* produce new integrals of motion. The proof exploits the infinite Taylor expansion of non-polynomial analytic functions to show that group-averaged gradients span the same space as unaveraged gradients.

2. Positive result (Theorem 4): For MSE loss and *tensorizable networks* — a newly introduced class of architectures where input and parameter dependence separate via a lifted feature space — data augmentation *can* create new conserved quantities. The mechanism works through a lifted symmetry group H\mathcal{H} acting on the feature space, which can be continuous even when the data symmetry group GG is discrete.

The concept of tensorizable networks is itself a notable conceptual contribution. These are networks satisfying fθ(x)=M(θ)T(x)f_\theta(x) = M(\theta)T(x), encompassing linear networks, polynomial networks, and Lightning Attention. This abstraction cleanly separates where symmetry analysis can be performed.

Methodological Rigor

The mathematical framework is rigorous and carefully constructed. The proofs leverage sophisticated tools:

  • Theorem 3 uses a Vandermonde-type argument: the infinite non-zero Taylor coefficients of the analytic loss create an overdetermined system that prevents gradient collapse under group averaging. Assumption 1 (injectivity modulo signs of χx,θ\chi_{x,\theta}) is explicitly stated and shown to hold generically.
  • Theorem 4 relies on the first fundamental theorem of the orthogonal group (Weyl) to show that O(V)O(V)-invariance of the loss implies dependence only through P⊤PP^\top P, forcing the gradient flow to preserve range(P)\text{range}(P).
  • Appendix D provides a complete characterization of H\mathcal{H} using real representation theory (Maschke's theorem, Schur's lemma, Frobenius's classification of real division algebras), showing H\mathcal{H} decomposes into products of orthogonal, unitary, or compact symplectic groups depending on the type (real/complex/quaternionic) of irreducible representations.

    • The experiments are explicitly presented as qualitative/illustrative rather than empirical validation at scale, which is appropriate for a theory paper. They demonstrate approximate conservation under gradient descent discretization and finite Haar sampling.

    Potential Impact

    Theoretical impact: This work opens a new direction in understanding how data structure interacts with optimization geometry. The classical conservation law literature for neural networks (Marcotte et al. 2023, 2024) focused on data-independent integrals of motion. This paper extends that framework to data-dependent settings, which is more realistic.

    Practical implications: Understanding conserved quantities constrains the optimization landscape, which has downstream consequences for:

  • Implicit bias characterization (what solutions gradient descent favors)
  • Initialization sensitivity analysis
  • Understanding when data augmentation changes optimization dynamics qualitatively versus merely smoothing the loss

    • Connections to adjacent fields: The Noether-inspired framework bridges classical mechanics, representation theory, and deep learning theory. The characterization of H\mathcal{H} via Frobenius's theorem and the appearance of orthogonal/unitary/symplectic groups suggests deep structural connections to physics-inspired machine learning.

    Timeliness & Relevance

    The paper is well-timed. There is growing interest in (1) geometric deep learning and equivariant architectures, (2) implicit bias and conservation laws in optimization, and (3) understanding data augmentation theoretically. This work sits precisely at the intersection, providing a principled framework where these threads meet.

    The inclusion of Lightning Attention as a tensorizable network is particularly timely given the dominance of attention mechanisms, though the analysis applies to a simplified (unnormalized, single-head) variant.

    Strengths

  • Clean dichotomy: The contrast between analytic non-polynomial losses (no new conservation laws) and polynomial/MSE loss (possible new conservation laws) is elegant and provides clear conceptual guidance.
  • Complete algebraic characterization: The full characterization of H\mathcal{H} in Appendix D via representation theory is thorough and self-contained.
  • Novel architectural abstraction: Tensorizable networks provide a useful conceptual tool that may find applications beyond this paper.
  • Concrete examples: The C3C_3-linear model and Lightning Attention examples make the abstract framework tangible.

    • Limitations

  • Restricted loss functions: Theorem 3 covers only margin losses; cross-entropy with softmax, for instance, is not directly addressed. Theorem 4 requires MSE specifically.
  • Tensorizable networks are restrictive: Many practical architectures (ReLU networks, transformers with softmax attention, normalization layers) are not tensorizable. The paper acknowledges this but leaves extensions to future work.
  • Realizability gap: Not every H\mathcal{H}-symmetry can be pulled back to parameter space via condition (25). The paper does not characterize when realizability holds in general.
  • No discussion of approximate conservation: In practice, approximate symmetries and finite training time mean exact conservation is never achieved. The framework does not address stability or perturbation analysis.
  • Scale of experiments: The experiments are minimal (3D linear model, small attention model). While appropriate for a theory paper, larger-scale verification would strengthen confidence.
  • Infinite groups: Theorem 3 is restricted to finite groups; whether similar results hold for compact infinite groups is unaddressed.

    • Overall Assessment

    This is a mathematically sophisticated paper that introduces a well-motivated question and provides clean, rigorous answers under specific assumptions. The dichotomy between polynomial and non-polynomial losses is a genuine insight. The tensorizable network framework and the complete representation-theoretic characterization of H\mathcal{H} represent substantial intellectual contributions. However, the practical applicability is currently limited by the restrictiveness of both the loss function assumptions and the tensorizable architecture class.

    Rating:6.8/ 10
    Significance 7Rigor 8.5Novelty 7.5Clarity 7.5

    Generated Jun 10, 2026

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