General $E(2)$-Equivariant Steerable CNNs

• The paper gives a general strategy that reduces kernel constraints under arbitrary representations to simpler constraints under individual irreducible representations.It also provides a general solution for the Euclidean group E(2) and its subgroups.Source: §1 · p. 1
• The resulting unified framework implements regular GCNNs [6] [7] [9] [10] [11], classical Steerable CNNs, Harmonic Networks [12], and several other existing and new architectures.It supports hybrid models that mix different representation types across and within layers.Source: §1 · p. 1
• A group restriction operation enables architectures whose equivariance decreases with depth, matching natural images whose low-level features are broadly oriented but globally orientation-biased.This adaptive equivariance accounts for symmetry loss through the feature hierarchy.Source: §1 · p. 2
• A drop-in replacement with equivariant convolutions yields significant gains over non-equivariant baselines on CIFAR10, CIFAR100, and STL-10.The benchmark also compares groups, representations, and nonlinearities on MNIST 12k, rotated MNIST SO(2), and reflected and rotated MNIST O(2).Source: §1 · p. 2
• The kernel-constraint results extend to steerable CNNs on homogeneous spaces [3] [4], gauge-equivariant CNNs on manifolds [5], spherical CNNs [5] [20] [21] [22] [23], and broader geometric deep learning models [24] [25].Two-dimensional manifolds with orthogonal structure group O(2), or its subgroups, require the same kernel constraints.Source: §1 · p. 2

2.1 Isometries of the Euclidean plane R2

• E(2) models planar translations, rotations, and reflections, with subgroups providing continuous or discrete levels of equivariance for oriented and unoriented image features.The framework considers groups of the form (R2, +) ⋊ G, including SO(2), CN, DN, axis reflections, and O(2).Source: §2.1 · pp. 2–3

2.2 E(2) - steerable feature fields

• Steerable feature fields assign each spatial feature vector a representation ρ that specifies channel mixing under group actions, with scalar and vector fields as key examples.Stacks of fields transform under direct-sum representations, while intermediate representation types are user-selected hyperparameters.Source: §2.2 · p. 3

2.3 E(2) - steerable convolutions

• Equivariant linear maps are precisely convolutions with G-steerable kernels satisfying a linear constraint, whose lower-dimensional solution space improves parameter efficiency over unconstrained kernels.The constraint ensures outputs transform according to the prescribed output representation when inputs transform according to the input representation.Source: §2.3 · p. 4

2.4 Irrep decomposition of the kernel constraint

• The arbitrary-representation kernel constraint decomposes into independent constraints on irreducible-representation blocks, enabling complete kernel bases to be constructed blockwise.The strategy is computationally more efficient for large representations and applies beyond subgroups of O(2) to steerable CNNs generally.Source: §2.4 · pp. 4–5

2.5 General solution of the kernel constraint for O(2) and subgroups

• For O(2) and its subgroups, norm preservation leaves radial kernel components free while Fourier analysis determines admissible angular harmonics and yields complete steerable-kernel bases.Subgroups G < O(2) impose weaker constraints and therefore permit higher-dimensional kernel bases, whereas stronger equivariance improves parameter efficiency.Source: §2.5 · p. 5, Table 2 · p. 5

2.6 Group representations and nonlinearities

• Equivalent representations yield equivalent convolution bases, so representation design reduces to selecting irreducible-representation types and multiplicities while accounting for nonlinear-layer compatibility.Regular and quotient representations trade channel count against symmetry restrictions: quotient fields use |H| fewer channels but restrict the steerable kernel basis.Source: §2.6 · pp. 5–6
• Representation choices constrain admissible pointwise nonlinearities: unitary fields support norm-based nonlinearities, permutation fields support pointwise nonlinearities, and induced representations inherit nonlinearities from subgroup representations.Norm-ReLUs, squashing, gated nonlinearities, regular representations, quotient representations, pooling, vector-field nonlinearities, and tensor products are discussed.Source: §2.6 · pp. 5–6
• Concatenated feature fields transform under direct-sum representations, allowing each constituent field to use its corresponding equivariant nonlinearity, while steerable-CNN theory does not privilege particular representations or nonlinearities.Different representation and nonlinearity choices are therefore compared empirically in Section 3.1.Source: §2.6 · p. 7

2.7 Group restrictions and inductions

• Equivariance can be reduced across layers by restricting a G-representation to a subgroup H, while local H-patterns can be lifted to larger-scale G-equivariant fields through induction.This supports scale-dependent symmetry, such as retaining reflections after removing rotations, and subsequent H-steerable convolutions can process the restricted fields.Source: §2.7 · p. 7

2.8 Implementation details

• The implementation precomputes representation-dependent steerable-kernel bases, expands kernels with learned coefficients, samples them on grids with aliasing-aware frequency cutoffs, and applies standard convolution routines.Block-diagonal representations decompose the kernel constraint into independent field-pair blocks, improving computational efficiency; the implementation is available as the e2cnn PyTorch extension.Source: §2.8 · pp. 7–8
• Grid sampling makes equivariance approximate for groups that are not symmetries of the pixel lattice, with exact Z2 symmetries restricted to subgroups of D4.This limitation motivates using grid-compatible discrete symmetry groups in related work.Source: §2.8 · p. 8
• At test time, E(2)-steerable CNNs incur no computational overhead over conventional CNNs because expanded kernels can be reused in evaluation mode.Kernel expansion is outweighed by convolution cost for reasonably sized images, and parameters are not updated during evaluation.Source: §2.8 · p. 8

3.1 Model benchmarking on transformed MNIST datasets

• On transformed MNIST, regular steerable models outperform the CNN baseline, with accuracy saturating near 8–12 orientations and DN models exceeding CN models on MNIST O(2).Restricted models exploit local symmetries while retaining global orientation information, whereas quotient representations show no significant difference from regular representations.Source: §3.1 · pp. 10, 12, Figure 2 · p. 10, Table 3 · p. 11
• Gated nonlinearities produce the best MNIST benchmark results, while induced O(2) representations outperform pure O(2) irreps but remain below large-order DN models.Pure SO(2) and O(2) irrep models are generally less competitive, and isotropic scalar-only kernels can perform worse than conventional CNNs.Source: §3.1 · pp. 12–13

3.2 MNIST group restriction experiments

• Restricting equivariance later in the network improves MNIST performance because models retain local symmetries while becoming globally invariant only to transformations present in the data.All restricted models outperform globally invariant, non-restricted models.Source: Table 4 · p. 13, §3.2 · p. 14

3.3 On the convergence of Steerable CNNs

• Steerable CNNs converge significantly faster than conventional CNNs, with convergence improving as rotation order increases and saturating around N = 8.Equivariance reduces intra-class variability that the network must learn, improving data efficiency and potentially reducing training time.Source: §3.3 · p. 14, Figure 3 · p. 14

3.4 Competitive MNIST rot experiments

• A D16-equivariant MNIST rot model restricted to C16 after the penultimate layer establishes a new state of the art, while quotient representations slightly improve the replicated C16 model.The replicated C16 model matches the original previous-SOTA accuracy before these modifications.Source: §3.4 · p. 14, Table 5 · p. 15

3.5 CIFAR experiments

• Without hyperparameter tuning, G-steerable WideResNets significantly outperform non-equivariant CIFAR baselines, including with AutoAugment; D8 D4 D4 gives the best non-augmented results.The computationally matched wrn28/10* D8 D4 D1 also substantially beats the baseline, showing benefits beyond increased computation.Source: §3.5 · p. 15, Table 6 · p. 15

3.6 STL-10 experiments

• On STL-10, every equivariant model significantly improves over the baseline, while the small D8 D4 D1 model outperforms the large D1 D1 D1 model.Equivariance gains persist across training-set sizes, and rotational equivariance is especially beneficial for the higher-resolution images.Source: Table 7 · p. 15, §3.6 · p. 16

4 Conclusions

• Equivariant models retain significant improvements across all STL-10 dataset sizes in the data-ablation study.Figure 4 reports this trend for the evaluated dataset sizes.Source: Figure 4 · p. 16
• G-steerable convolutions significantly improve CIFAR-10, CIFAR-100, and STL-10 accuracy without additional hyperparameter tuning.Even non-scaled equivariant models outperform conventional CNNs, while faster convergence can reduce training cost.Source: §4 · p. 16
• The theory unifies many rotation- and reflection-equivariant models by analytically solving kernel constraints for representations of O(2) and its subgroups.This framework enables systematic reproduction and comparison of existing models.Source: §4 · p. 16
• Group restriction adapts equivariance to the symmetries present at each spatial length scale.The operation is proposed as a way to match model equivariance to scale-dependent symmetries.Source: §4 · p. 16
• Steerable CNNs incur a small training-time kernel-expansion overhead but can match conventional CNN test-time cost at the same width.Their parameter efficiency also motivates width adjustment when matching conventional-model parameter budgets.Source: §4 · p. 16
• The authors identify biomedical imaging and broader vision tasks with global or local symmetries as promising applications for E(2)-steerable CNNs.They note that generalization across settings and the wide range of steerable-CNN design choices still require further investigation, while their library is intended to facilitate adoption and research.Source: §4 · p. 16

A Local gauge equivariance of E(2)-steerable CNNs

• The original derivation handles active translations, rotations, and reflections, but does not itself establish equivariance to independently transformed local patches.The active action moves signal values as g^-1(x − t), whereas local transformations require the passive gauge framework [2] [3] [4].Source: A Local gauge equivariance of… · p. 17
• The E(2)-steerable CNN kernel constraint guarantees equivariance to independent local rotations and reflections of image patches, not only global transformations.Gauge transformations act independently at each position, and their required kernel constraint is identical to Eq. (2) [5].Source: A Local gauge equivariance of… · p. 17, Figure 5 · p. 17
• The same kernel constraint applies on arbitrary two-dimensional Riemannian manifolds, covering spherical CNNs and convolutional networks on triangulated meshes.Different structure groups G ≤ O(2) and group representations yield the corresponding architectures [5] [20] [21] [22] [23] [24] [25].Source: A Local gauge equivariance of… · p. 17

B A short primer on group representation theory

• A linear representation is a homomorphism ρ: G → GL(R^n) satisfying ρ(g1g2) = ρ(g1)ρ(g2), thereby encoding feature-space transformation laws.The homomorphism condition makes matrix multiplication compatible with group composition, although representations need not be faithful.Source: B A short primer on group… · p. 18
• Representations with no non-trivial invariant subspace are irreducible, and compact-group representations can be decomposed into irreducible components after a change of basis.This reduction lets proofs of the kernel constraint consider irreducible representations [3].Source: B A short primer on group… · p. 18
• Regular and quotient representations act by permuting group elements or cosets, with regular and trivial representations arising as quotient representations for H = {e} and H = G.Vectors can be viewed as scalar functions on G or G/H, whose actions correspond to left translations.Source: B A short primer on group… · pp. 18–19
• An induced representation combines coset permutation with the subgroup representation on each coset-associated subspace, generalizing to non-finite groups.For G = E(2) = (R^2, +) ⋊ O(2), H = O(2), and G/H = R^2, the induced action corresponds to Eq. (1) [3].Source: B A short primer on group… · pp. 19–20

C An intuition for quotient representation fields

• Quotient representations store only N/M non-redundant responses for CM-symmetric patterns, reducing computational, memory, and parameter costs while retaining selected orientation information.Regular fields redundantly repeat responses after N/M orientations; mixed regular, quotient, and trivial representations slightly improve over an equally parameterized regular model on MNIST, but incorrect assumed symmetries can severely harm performance.Source: C An intuition for quotient… · p. 21

D.1 Equivariance of E(2) - steerable convolutions

• G-steerable convolution kernels guarantee equivariance of the convolution mapping under Euclidean-group actions.The derivation uses the kernel steerability constraint and an invariant change of integration variables; necessity is established in [2].Source: §D.1 · p. 22

D.2 Equivariance of spatially localized nonlinearities

• Spatially localized nonlinearities are equivariant when their feature-vector maps commute with the corresponding input and output representations.This framework covers direct-sum, norm, pointwise regular or quotient, vector-field, and induced-representation nonlinearities.Source: §D.2 · p. 22, §D.2.1 · p. 23, §D.2.2 · p. 23

D.2.3 Equivariance of pointwise nonlinearities w.r.t. regular and quotient representations

• Pointwise nonlinearities remain equivariant for regular and quotient representations because these representations permute feature entries, while vector-field nonlinearities use the maximum response to produce an equivariant direction.For vector fields, the resulting feature field transforms like a vector field under cyclic-group rotations.Source: §D.2.3 · pp. 23–24, §D.2.4 · p. 24

E Visualizations of the irrep kernel constraint

• The irrep kernel constraint decomposes into independent constraints on kernel blocks associated with input-output irreducible subspaces.Each block contributes to the basis of full G-steerable kernels through zero-padded assembly.Source: E Visualizations of the irrep kernel… · p. 25

F Solutions of the kernel constraints for irreducible representations

• Analytical solutions are provided for every pair of irreps of SO(2), O(2), reflection, CN, and DN in Tables 8–12.The solutions form a linear subspace of unrestricted kernels and are derived using real-valued irreducible representations.Source: F Solutions of the kernel constraints… · p. 26
• Because the irreps are angular harmonics, kernels are expanded in polar Fourier-like bases, and allowed expansion coefficients parameterize the complete steerable-kernel space.Completeness follows from completeness of the orthogonal basis; discrete rotations additionally yield linear congruences modulo N.Source: F Solutions of the kernel constraints… · p. 26

F.1 Conventions, notation and basic properties

• The representation conventions cover rotations and reflections as semidirect-product elements, while real irreps include frequency-limited SO(2), O(2), and cyclic-group cases.For CN, frequencies above floor(N/2) are aliased because rotation angles are discrete.Source: §F.1 · p. 26, §F.2 · p. 27

F.2 Irreducible representations of G≤O(2)

• F.2 describes subgroup irreps, including frequencies up to floor(N/2) for dihedral groups and two one-dimensional irreps when N is even.The reflection group ({±1}, ∗) is isomorphic to D1 and C2 and therefore has the same irreps as those groups.Source: §F.2 · p. 28

F.3 Analytical solutions of the irrep kernel constraints

• F.3 provides angular bases satisfying the irrep kernel constraint for SO(2), O(2), reflection, cyclic, and dihedral groups.The bases are tabulated for pairs of input and output irreps; cyclic and dihedral bases are completed by instantiating solutions over t∈Z or t̂∈N.Source: Table 8 · p. 29, Table 9 · p. 29, Table 10 · p. 29
• F.3 assumes reflection about the horizontal axis, β=0; for an axis at β≠0, the angular variable is replaced by φ−β.The dihedral solutions use the same axis convention.Source: §F.3 · p. 30

F.4 Derivations of the kernel constraints

• Kernel constraints are solved for 1- and 2-dimensional irreducible representations of G ≤ O(2), covering all dimension-mapping cases.The cases are mappings between 2-dimensional irreps, between 2- and 1-dimensional irreps in either direction, and between 1-dimensional irreps.Source: §F.4 · p. 31
• The derivations initially consider only positive radial parts r > 0, while the origin r = 0 requires additional considerations deferred to Section F.4.6.Source: §F.4 · p. 31

Derivation

• The derivation restricts attention to angular kernels on S1 by dropping the unrestricted radial component, then solves the constraint through projections onto Fourier-derived basis elements.Orthogonality of the angular basis makes mismatched frequencies integrate to zero, reducing the constraint to frequency conditions.Source: §F.4.1 · pp. 32, 34

2-dimensional irreps

• For m,n > 0, the SO(2)-steerable angular kernel space has a four-dimensional basis constrained by μ′ = m − sn′, with both γ and s unrestricted.The basis is explicitly given in the lower-right cell of Table 8.Source: §F.4.1 · p. 34

1-dimensional irreps

• For trivial representations m = n = 0, rotational invariance eliminates all nonzero frequencies and leaves a one-dimensional basis of isotropic kernels.This basis appears in the upper-left cell of Table 8.Source: §F.4.1 · pp. 34–36

1 and 2-dimensional irreps

• For a 1-dimensional input and 2-dimensional output, projection enforces μ′ = m and produces a two-dimensional equivariant-kernel basis for m > 0 and n = 0.The basis is explicitly given in the lower-left cell of Table 8.Source: §F.4.1 · p. 37

2 and 1-dimensional irreps

• For a 2-dimensional input and 1-dimensional output, the derivation is identical up to transposition and yields a two-dimensional basis for m = 0 and n > 0.The resulting basis is shown in the upper-right cell of Table 8.Source: §F.4.1 · p. 37

F.4.2 Derivation for the reflection group

• The reflection-group calculation yields a one-dimensional kernel for the two one-dimensional irreducible representations, with the zero-frequency coefficient vanishing when their representation labels differ.For positive frequencies, the surviving phase is γ′ = (i + j mod 2)π/2.Source: §F.4.2 · pp. 38–39
• Only half of the β = 0 Fourier basis is preserved by reflection equivariance, with the general-axis solution recovered by reorienting the kernel.The reflection constraint depends on the axis angle β, but a transformed kernel reduces it to the β = 0 case; solutions for other axes follow by reversing this transformation.Source: §F.4.2 · pp. 38–39

F.4.3 Derivation for O(2)

• O(2) is the semidirect product of SO(2) and the reflection group, so its equivariant kernel space is the intersection of the rotation- and reflection-equivariant spaces.This decomposition justifies imposing both constraints rather than deriving an independent O(2) constraint.Source: §F.4.3 · p. 40

Proof

• The proof establishes both directions of the equivalence: satisfying SO(2) and reflection constraints implies O(2) equivariance, while O(2) equivariance trivially implies each constituent constraint.Thus the intersection construction is exact, not merely sufficient.Source: Proof · pp. 40–41
• For general representations, the O(2) basis is constructed by restricting representations to SO(2) and reflections, computing both bases, and intersecting them.For two-dimensional representations, the reflection basis retains the γ = 0 subset of the SO(2) Fourier basis while also imposing the SO(2) frequency condition.Source: Proof · pp. 41–42
• When restricted representations decompose into one-dimensional reflection irreducibles, the kernel constraint separates into independent entrywise constraints that can be solved and recombined.This provides the construction for mixed one- and two-dimensional representation cases.Source: Proof · pp. 41, 43
• For zero-frequency one-dimensional cases, intersecting reflection and rotation conditions enforces equal representation labels, yielding the corresponding O(2)-equivariant basis.The reflection condition selects γ = (i + j mod 2)π/2, whereas the SO(2) condition requires μ = 0 and γ = 0.Source: Proof · p. 42
• The remaining mixed representation cases follow the same intersection procedure, producing one-dimensional O(2)-equivariant bases and transpose-related bases for reversed input-output dimensions.These cases are organized in the corresponding cells of Table 9.Source: Proof · p. 43

General derivation

• CN reuses the SO(2) Fourier basis and projection operators, while enforcing projected constraints only for discrete angles.The kernel-constraint projection on the left-hand side is unchanged from SO(2).Source: §F.4.4 · pp. 43–44

1-dimensional irreps

• When N is even, CN has an additional one-dimensional irrep with frequency N/2; otherwise this irrep is absent.The one-dimensional cases are therefore rederived using frequencies in {0, N/2}.Source: §F.4.4 · pp. 43–44

2-dimensional irreps

• For two-dimensional irreps, the CN basis follows the SO(2) basis but also includes solutions for aliased frequencies obtained by adding tN.The discrete constraints reduce to congruence conditions modulo N, with the p = 1 constraint sufficient for the remaining values of p.Source: §F.4.4 · pp. 44, 46

1-dimensional irreps

• For one-dimensional input and output irreps, valid frequencies satisfy µ′ = (m+n mod N) + tN, yielding the corresponding basis in Table 11.The constraint is required for every discrete angle, and the p = 1 case captures all solutions.Source: §F.4.4 · p. 46

Mixed irreps

• For mixed one- and two-dimensional representations, the derived basis is obtained from discrete frequency-matching constraints, while reversing input and output gives the transpose basis.The one-dimensional-input/two-dimensional-output case appears in Table 11’s bottom-left cells; the reverse case is identical up to transpose.Source: §F.4.4 · pp. 46–47

F.4.5 Derivation for DN

• DN-equivariant kernels are derived by repeating the O(2) procedure with the CN bases as the starting point.The resulting solutions are organized in Table 12.Source: §F.4.5 · p. 48
• Because DN is not transitive on reflection axes, the axis β matters; derivations may use β = 0 and recover general axes by κ′(r, φ) = κ(r, φ − β).Reflection axes are equivalent only when β = β′ + n 2π/N for some integer 0 ≤ n < N.Source: §F.4.5 · p. 48

F.4.6 Kernel constraints at the origin

• At the origin, polar-coordinate ambiguity requires angularly constant kernels, restricting the Fourier bases to zero frequencies μ = 0.Apart from this restriction, the kernel constraints at r = 0 match those for r > 0.Source: §F.4.6 · p. 49

F.5 Complex valued representations and Harmonic Networks

• Complex representations simplify SO(2) and CN derivations because their irreducible representations are 1-dimensional, but converting solutions to real implementations can underparameterize kernels.For O(2) and DN, complex and real irreducible representations are equivalent up to a basis change.Source: §F.5 · p. 49
• Harmonic Networks [12] use a 2-dimensional real kernel space where the corresponding SO(2)-steerable basis here is 4-dimensional, omitting solutions with frequency m + n.This comparison concerns interpreting the real feature fields as transforming under real irreducible representations.Source: §F.5 · p. 50
• The dimensionality gap arises because real SO(2) irreducible representations decompose into complex irreducible representations with opposite frequencies, whose real intertwiners supply the additional solutions.Positive and negative complex frequencies are conjugate and equivalent up to a change of basis.Source: §F.5 · p. 50
• Harmonic Networks [12] are complete over complex fields, while real-field implementation implicitly adds negative frequencies and enables the larger steerable basis without overhead.Thus the real implementation can use the broader basis without adding computational overhead.Source: §F.5 · p. 50

Numerical approach

• The numerical algorithm in [2] generalizes to arbitrary unitary representations by solving a stacked Sylvester equation and extracting its null space.It needs only the generators of the group, whose solutions automatically extend to all group elements.Source: G Alternative approaches to compute… · p. 51
• Although general, the method in [2] becomes extremely expensive for large representations because SVD-based null-space computation and basis changes scale poorly with representation dimensions.Its unfavorable complexity was negligible in [2] because that work used relatively small irreducible representations.Source: G Alternative approaches to compute… · p. 51

Proposed approach

• The proposed algorithm independently decomposes input and output representations into irreducibles and applies analytically derived kernel solutions for subgroups G ≤ O(2).Independent input/output changes of basis reduce the computational burden compared with solving one large numerical decomposition.Source: G Alternative approaches to compute… · p. 51
• The improved implementation makes large representations feasible, including the representation with dim(ρD20 reg) = 40 used in this work.This contrasts with the numerical approach’s poor scaling on large representations.Source: G Alternative approaches to compute… · p. 51

MNIST benchmarks

• The MNIST benchmark models derive from a basic architecture whose convolution blocks combine convolution, batch normalization, and a nonlinearity.The first fully connected layer is followed by batch normalization and ELU.Source: Table 13 · p. 52

MNISTrot experiments

• The final MNISTrot architecture uses dropout with p = 0.3 before each fully connected layer, with batch normalization and ELU after the first two.This architecture is replicated from [7].Source: Table 14 · p. 52

H.1 Benchmarking on transformed MNIST datasets

• All reported models match parameter counts and relative layer widths, use Adam with batch size 64, and select the best validation checkpoint after 30 epochs.Learning rates start at 10^-3, decay by 0.8 per epoch after 10 epochs, and convolutional and first fully connected layers use weight decay 10^-7.Source: §H.1 · p. 52
• MNIST steerable bases use Gaussian radial widths σ = 0.6 except σ = 0.4 on the outermost ring, with strong frequency cutoffs to limit equivariance errors.The cutoffs permit frequencies up to 0, 2, 2 for radii 0, 1, 2 in 5×5 kernels and 0, 2, 3, 2 for radii 0, 1, 2, 3 in 7×7 kernels; continuous rotations and reflections augment applicable datasets.Source: §H.1 · p. 52

H.2 Competitive runs on MNIST rot

• Competitive MNIST rot experiments replicate the state-of-the-art C16-equivariant architecture of [7], while using different frequency bandlimits and Gaussian radial widths.Layer widths are reported as numbers of fields in the C16 regular model.Source: §H.2 · p. 52
• Final MNIST rot models train on 10000 + 2000 samples for 40 epochs with Adam, batch size 64, train-time augmentation, and dropout p = 0.3 in fully connected layers.The initial learning rate is 0.015, decayed by 0.8 after 15 epochs, with L1 and L2 regularization weight 10^-7.Source: §H.2 · p. 52

H.3 CIFAR experiments

• CIFAR models replace WideResNet convolutions with G-steerable convolutions while preserving total parameter counts and map final features to scalar fields for invariant predictions.D8- and C8-equivariant blocks use 5×5 kernels, and all rings use Gaussian width σ = 0.45.Source: §H.3 · p. 53
• CIFAR steerable models use higher frequency cutoffs, such as 0, 3, 6 for radii 0, 1, 2, because natural-image classification tolerates the resulting small discretization errors.Training follows WideResNet practice for 200 epochs with SGD, batch size 128, learning rate 0.1, momentum 0.9, weight decay 5 · 10^-4, and standard crop, flip, and normalization augmentation.Source: §H.3 · p. 53

Experimental setup

• The experiments correct [43] using GitHub-reported settings, including 60 × 60 cutout patches, stride 2 in the first block’s initial convolution, and CIFAR-10 normalization.The CIFAR-10 statistics are close to STL-10’s but may affect performance because STL-10 has a small labeled training set.Source: §H.4 · p. 53

Model architecture

• Regular feature fields are used throughout except in the last convolution, which maps to trivial fields; in the small model, D8’s 16-channel regular representation limits the first convolution to one independent filter repeated 16 times.The first convolution’s output channels are increased by the square root of the group size.Source: §H.4 · p. 53
• The kernel basis uses ring width σ = 0.6, outermost-ring width σ = 0.4, and frequency cutoffs 0, 3, 6, … determined by a factor of 3.These settings define the radial rings and their frequency limits for the steerable kernels.Source: §H.4 · p. 53

Training procedure

• Training uses only labeled images for 1000 epochs with batch size 128, SGD with Nesterov momentum 0.9, weight decay 5 · 10^-4, and scheduled learning-rate reductions.The learning rate starts at 0.1 and is divided by 5 at epochs 300, 400, 600, and 800; augmentation uses padding, random 96 × 96 crops, horizontal mirroring, and CutOut.Source: §H.4 · p. 53

Data ablation

• The data ablation mixes the original train, validation, and test sets into balanced datasets of 250, 500, 1000, 2000, or 4000 training images, averaging four runs per size.Validation and test sets contain 1000 and 8000 images and are resampled in each run; the same models and training procedure as the full-data experiment are used.Source: §H.4 · p. 53

SO(2) models

• The best SO(2) variant applies shared norm-ReLUs to multiple non-trivial irrep fields and computes norms of composed fields, significantly outperforming earlier variants across all datasets.Merging scalar fields into combined fields performs significantly worse, while combining invariant norm mappings with norm-ReLUs does not improve on the baseline.Source: I Additional information on the irrep… · p. 54
• For SO(2) irrep models, gated nonlinearities perform significantly better than every other tested nonlinearity, while norm-based invariant maps consistently beat conv2triv.Sharing gates across non-trivial fields does not significantly affect performance.Source: I Additional information on the irrep… · p. 54

O(2) models

• O(2)-conv2triv maps to scalar and sign-flip fields, takes absolute values of sign-flip outputs, and avoids direct trivial mapping because the relevant kernel space would produce dead neurons.The resulting invariant features are multiple scalar fields formed from f0,0 and |f1,0|.Source: I Additional information on the irrep… · p. 54
• For induced O(2) fields, Ind-conv2triv and Ind-norm use G-pooling, while Ind norm-ReLU applies equal-bias subfield nonlinearities and Ind gate uses equivariantly permuting gates computed from IndO(2) 0-fields.These designs account for the permutation of subfields under reflections.Source: I Additional information on the irrep… · p. 55
• IndO(2) SO(2) models perform much better than pure irrep models despite being equivalent up to a change of basis.Induced representations disentangle reflections as permutations, unlike direct sums of irreps, paralleling the advantage of regular representations over corresponding irrep sums.Source: I Additional information on the irrep… · p. 55