Automatic Symmetry Discovery with Lie Algebra Convolutional Network

1 Introduction

• Most equivariant-network approaches require the symmetry group in advance, making model selection difficult when the symmetries are unknown.Many existing methods also focus on finite groups.Source: §1 · p. 1
• For continuous groups, existing methods discretize the group or sum over irreducible representations, approaches described as prone to approximation error.Other methods use Monte Carlo sampling or require explicit group information and constraint solving.Source: §1 · p. 2
• L-conv uses Lie algebra generators to encode groups without discretization or summing over irreducible representations, and its basis can be learned to discover symmetries.The authors also state that it uses fewer parameters than Zhou et al. (2020) and is more general than Augerino, which is restricted to a subgroup of 2D affine transformations.Source: §1 · p. 2
• The authors present L-conv as a building block for group-equivariant networks, show that multilayer L-conv approximates group convolution including CNNs, and identify graph convolutional networks as a special case.They also connect equivariant loss, robustness, and equivariance to Lagrangians, Euler-Lagrange equations, and Noether currents.Source: §1 · p. 2

2 Background

• A group action on the base space induces an action on feature maps, and a mapping is equivariant when transforming its input corresponds to transforming its output.The background defines scalar feature transformations and maps between feature spaces.Source: §2 · p. 2
• A linear equivariant map can be expressed as group convolution, which lifts base-space points to group elements and applies a kernel over the group.The group’s Lie algebra basis can be pushed forward to vector fields on the base space.Source: §2 · pp. 2–4
• The paper uses infinitesimal generators rather than discretizing a group or summing over irreducible representations to build an approximation to group convolution.For matrix representations, the Lie bracket is the commutator of generator matrices.Source: §2 · p. 3
• A Lie group’s Lie algebra is its tangent space at the identity, with basis elements called infinitesimal generators.For connected groups, elements can be reached by paths from the identity and represented using a path-ordered exponential.Source: §2 · p. 3, Figure 1 · p. 3

3 Lie Algebra Convolutional Network

• A kernel concentrated near the identity can be expanded in the Lie algebra, allowing its effect to be represented by weights and infinitesimal-generator derivatives instead of an explicit group integral.The generators act as vector fields on the base space; translations yield partial derivatives, while 2D rotations yield the angular derivative.Source: §3 · pp. 4–5

3.1 Approximating G-conv using L-conv

• Figure 2’s procedure approximates a group-convolution kernel by summing localized kernels and moving each toward the identity through multiple L-conv layers.The kernel approximation uses small neighborhoods around selected group elements.Source: Figure 2 · p. 5, §3.1 · pp. 5–6

3 Lie Algebra Convolutional Network

• L-conv is equivariant under the group, as established by Proposition 1.The paper illustrates generator operators for translations, rotations, and rotations with scaling.Source: §3 · p. 5

3.1 Approximating G-conv using L-conv

• Theorem 1 states that multilayer L-conv approximates group convolution, and for compact groups the authors extend this to equivariant feedforward networks with pointwise nonlinearities.The corollary assumes the group acts transitively on the base space.Source: §3.1 · pp. 5–6
• For continuously differentiable kernels with compact support, localized-kernel approximation has an error bound that can be made arbitrarily small by choosing sufficiently small neighborhoods.The stated bound is |G0|η^2ξ^2, with η the neighborhood size and ξ bounding the kernel derivative.Source: §3.1 · p. 5
• The authors note that this result is not a universal approximation theorem for all equivariant scalar functions, because it is unknown whether every such function is approximable by an equivariant feedforward network.The stated result concerns the expressivity of multilayer L-conv relative to other equivariant networks.Source: §3.1 · p. 6

Tensor notation and implementation

• Each Lie-algebra operator can be implemented as a GCN module, with a residual feature transform and shared convolutional weights.The operators are aggregations, while the feature weights are ϵiW 0T.Source: §4 · p. 7, Figure 3 · p. 7
• The discrete-space generators have constrained structure: encoding the space as a graph makes the Lie-algebra terms edge weightings rather than arbitrary d × d matrices.The construction fixes a gauge by mapping neighbors of the origin to neighbors of each position.Source: §4 · p. 7
• L-conv acts on discrete spatial positions and feature dimensions separately, with Lie-algebra operators on space and trainable matrices on features.The feature map is a d × m tensor, and the lifted group action preserves equivariance.Source: §4 · pp. 6–7, Figure 3 · p. 7
• The number of generators is a hyperparameter independent of the number of discretized points, and may be over-complete without being problematic.For an n-dimensional discretized space, the passage suggests nL ∼ O(n2).Source: §4 · p. 7

Learning symmetries using L-conv

• In the SO(2) experiment, recursive L-conv learned a generator from image pairs, while the fixed-angle version was less noisy but omitted weights beyond first neighbors.The figure compares the recursive and fixed-angle generators with a numeric solution; the fixed angle was θ = π/10.Source: §4 · p. 7, Figure 4 · p. 8
• L-conv was tested only in limited experiments on small image datasets and may help find symmetries in scrambled data or data with hidden symmetries.The paper presents it as a potential replacement for other equivariant layers.Source: §4 · p. 7

CNN

• A 1D CNN can be approximated by multilayer L-conv because the translation generator acts as a convolution and can reproduce discrete shifts.The passage states that the argument generalizes to higher dimensions.Source: §5 · p. 8

Graph Convolutional Network (GCN)

• A GCN is a special case of L-conv when its Lie-algebra operator is a graph operator such as normalized adjacency.When all neighbors of each node have equal edge weight, the aggregation is uniform.Source: §5 · p. 8
• For a single generator h(A), the associated flow is g(t) = exp[h(A)t], and L-conv for such flow groups is a GCN.L-conv can instead assign different weights to neighbors of the same node, similar to GEM-CNN with a fixed gauge.Source: §5 · p. 8

Group invariant loss

• The loss can be made group-invariant by integrating over the group, and the formulation also permits integration over the base space after a change of variables.For supervised learning, inputs and labels are combined with a well-defined group action; categorical labels are unchanged by a continuous group.Source: §6 · p. 9

MSE loss and Field Theory

• The group-invariant MSE loss generalizes free field theories, with its induced tensor h acting as a Riemannian metric when features are scalar.For general feature spaces, h combines a spatial 2-tensor with an inner product on the feature space.Source: §6 · p. 9

Robustness and Euler-Lagrange Equation

• Robustness to small input perturbations requires the loss to remain unchanged around data points, yielding an Euler-Lagrange condition at optimal parameters.The passage frames robustness as δI[φ; W∗]/δφ = 0.Source: §6 · p. 9

Equivariance and Conservation laws

• When robustness error is minimized, deviations along a symmetry direction yield a divergence-free Noether current.The paper also proposes minimizing ∥∂αJα∥ as an alternative way to discover symmetry generators.Source: §6 · pp. 9–10
• The Noether current described here is the stress-energy tensor associated with space or space-time variations.The paper suggests that more general equivariant networks could produce other Noether currents.Source: §6 · p. 10

Conclusion and Discussions

• L-conv does not require irreps or group discretization and can approximate feedforward equivariant networks on compact groups.The conclusion also describes it as a simple, universal structure whose Lie-algebra basis can be learned.Source: §7 · p. 10
• A synthetic experiment validated that L-conv can learn the correct Lie-algebra basis.The paper connects equivariant networks to Noether’s theorem and conservation laws, and suggests extending the formulation to internal symmetries.Source: §7 · p. 10
• More expressive L-conv-based models may provide advanced ansatze for scientific problems, where simple perturbative Lagrangians can lead to divergences in quantum field theory.Source: §7 · p. 10

• The checklist says the code and datasets are proprietary and their license is not included.Source: Checklist · p. 12
• The authors report stating all theoretical assumptions and including complete proofs.Source: Checklist · p. 13
• The experiment checklist reports code, data, and reproduction instructions, training details, error bars, compute resources, and new assets as included.Source: Checklist · p. 13

A Extended derivations and proofs

• Equation 23 defines L-conv, whose equivariance is preserved when any following neural network acts only on feature indices, not spatial indices.Source: A Extended derivations and proofs · p. 15

A.1 Approximating G-conv using L-conv

• Multilayer L-conv approximates G-convolution, and under compactness and transitive action, approximates equivariant feedforward networks with pointwise nonlinearities.The text notes this is not a universal approximation theorem for all equivariant functions.Source: §A.1 · p. 16

A.2 Example of continuous L-conv

• Translations, rotations, and scaling are represented by generator flows, with the rotation generator acting as the angular derivative and the scaling generator as the radial derivative.Source: §A.2 · p. 17

A.3 Group invariant loss

• A group-invariant loss can be formed by integrating the squared equivariant output over the group, with labels transforming as scalar features or remaining categorical and fixed.Source: §A.3 · p. 18, §A.3.1 · p. 18

A.3.1 MSE Loss

• For scalar features, the loss-derived tensor h becomes a Riemannian metric; for translation symmetry, random Gaussian weights recover the Euclidean metric and the loss resembles a physics Lagrangian.Source: §A.3.1 · p. 19, §A.3.3 · p. 19

A.3.2 Third term as a boundary term

• The loss’s third term becomes a boundary term and generally vanishes when features have finite support and decay at infinity.Source: §A.3.2 · p. 19

A.3.4 Boundary term with spherical symmetry

• For spherical symmetry with scaling, the boundary contribution can remain nonzero only under radial, nonvanishing boundary features; such features imply divergent loss unless the mass matrix has zero eigenvalues.Source: §A.3.4 · p. 20

A.4 Robustness to random noise

• Robustness to small input perturbations is posed as stationarity of the loss around real data, and the resulting first variational term is the Euler–Lagrange equation.For finite-support features, the second term becomes a vanishing boundary integral, so robustness requires the data to satisfy the Euler–Lagrange equations.Source: §A.4 · p. 20
• For translation symmetry, the robustness equation becomes a Helmholtz equation.Source: §A.4 · p. 20

A.5 Conservation laws

• Infinitesimal equivariance of the loss integrand yields divergence-free terms identified as a Noether conserved current.The paper identifies the current with the stress-energy tensor associated with spatial or space-time variations.Source: §A.5 · pp. 20–21

B Tensor notation details

• In tensor notation, a group acts on discrete feature maps through matrix multiplication, and the rewritten L-conv expression is equivariant.Source: B Tensor notation details · pp. 21–22

B.1 Constraints from topology on tensor L-conv

• A graph incidence matrix encodes discrete topology, and each node’s outgoing-edge neighborhood approximates its tangent space through feature differences.The relation is exact for an n-dimensional square lattice in the infinitesimal-spacing limit.Source: §B.1 · pp. 22–23

B.2 Approximating a symmetry and discretization error

• When discretization resolution d greatly exceeds the Lie algebra dimension n, a subgroup of GL_d(R) can approximate the original continuous symmetry group, with approximation error depending on resolution.The authors assume the approximate group is the true symmetry because the original group cannot be measured directly from discretized data.Source: §B.2 · p. 23

B.3 Comparison with other symmetry discovery methods

• Compared with Zhou et al.’s meta-learned symmetry matrices, the paper says L-conv uses fewer parameters and has a direct Lie-algebra interpretation.Source: §B.3 · p. 24
• Unlike Augerino’s fixed affine generators and sampled transformations, L-conv learns generators directly without restricting to known symmetries or sampling the group.Source: §B.3 · p. 24

C Experiments

• The experiments learn generators directly with L-conv; for scrambled images, the generators are encoded as low-rank matrices.The broader symmetry-discovery discussion contrasts this approach with methods that sample group transformations or restrict generators to known symmetries.Source: C Experiments · pp. 24–25

C.1 Approximating 1D CNN

• L-conv approximates a 1D CNN because its Lie-algebra generator induces a convolution that can reproduce finite discrete shifts.The reported two-pixel shift approximations have cosine correlations of 0.77 for n = 8 and 0.93 for n = 16.Source: §C.1 · p. 25, Figure 6 · p. 25

C.2 Extracting 2D rotation generator for fixed small rotations

• Recursive L-conv learning of rotation angles yields 0.70 cosine correlation with the ground-truth rotation generator, compared with 0.27 for the small-angle estimate.The recursive method learns from image pairs differing by finite rotations; the fixed-angle experiment uses θ = π/10.Source: §C.2 · p. 26, Figure 7 · p. 26

C.3 Learning rotation angle

• For larger images and angles, angle prediction remains accurate (test loss 2.7 × 10^-4), while generator correlation falls to 0.12 with 8 recurrences.The learned generator is closer to a finite-angle rotation; the authors suggest this may better approximate larger rotations with few recurrences.Source: §C.3 · p. 27

D Experiments on Images

• Across four datasets, L-conv outperforms the tested baselines on rotated and rotated-and-scrambled images; on default CIFAR100, max-pooled CNN and two-layer SE(3) LieConv perform better.On default CIFAR100, L-conv remains ahead of the other tested models except those CNN and LieConv configurations.Source: Figure 9 · p. 27, Figure 10 · p. 28, D Experiments on Images · pp. 28–29
• Tests use MNIST, CIFAR10, CIFAR100, and FashionMNIST, with random rotations and fixed pixel permutations used to evaluate unfamiliar symmetries.The experiments use one- or two-hidden-layer models followed by classification, without dropout or batch normalization.Source: D Experiments on Images · pp. 27–28
• FC configurations perform worse than L-conv on CIFAR100, and random untrained generators cause a significant performance drop.The comparisons include parameter-matched shallow FC and LieConv; L-conv leads on rotated-and-scrambled CIFAR100.Source: D Experiments on Images · pp. 28–29

D.1 Details of experiments

• The L-conv layer uses resources comparable to CNN and runs only slightly slower in the reported implementation.The tests ran with TensorFlow 2.2 and Keras on a system with an NVIDIA Quadro P6000 GPU.Source: §D.1 · p. 29

D.2 Additional Experiments

• On rotated-and-scrambled MNIST, L-conv still outperforms CNN when CNN has the same or more parameters.The comparison varies kernel sizes to match model parameter counts.Source: §D.2 · p. 29
• Across four datasets, CNN performs best on default images while L-conv is second; on rotated and rotated-and-scrambled variants, L-conv performs best.MNIST FC and CNN results come close but use five times more parameters.Source: Figure 12 · p. 30
• On scrambled, rotated MNIST, most tested L-conv configurations achieve the highest accuracy without many trainable parameters, while fixed random generators perform worse.Figure 13 plots test accuracy against trainable parameter count; its grey lines show L-conv with fixed random generators.Source: §D.2 · p. 30, Figure 13 · p. 30