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A General Theory of Equivariant CNNs on Homogeneous Spaces
Taco Cohen, Mario Geiger, Maurice Weiler
TL;DR
G-CNN research lacks a unified way to relate existing equivariant layers and determine general constructions for new modalities. The paper models feature maps as fields on homogeneous spaces, classifies networks by symmetry group, base space, and field type, and proves that equivariant linear maps correspond to convolutions with equivariant kernels. The resulting formalism organizes existing G-CNNs and connects their layers to constrained kernel spaces.
Problem
Existing G-CNN approaches span several domains, but their relationships and the most general equivariant layers for new modalities are not immediately clear.
Method
The paper models feature spaces as fields on homogeneous spaces and characterizes equivariant linear maps through convolutions with kernels satisfying representation-dependent constraints.
Results
Equivariant linear maps between fields are in one-to-one correspondence with convolutions using equivariant kernels, whose spaces admit equivalent group, coset-space, and double-coset representations.
Takeaways & Limitations
The formalism provides a systematic language for classifying G-CNNs and supports the study and development of new equivariant network architectures.
Takeaways & Limitations
The theory covers fields over homogeneous spaces but not the more general setting of arbitrary manifolds.
Abstract
from arXiv · showhide
We present a general theory of Group equivariant Convolutional Neural Networks (G-CNNs) on homogeneous spaces such as Euclidean space and the sphere. Feature maps in these networks represent fields on a homogeneous base space, and layers are equivariant maps between spaces of fields. The theory enables a systematic classification of all existing G-CNNs in terms of their symmetry group, base space, and field type. We also consider a fundamental question: what is the most general kind of equivariant linear map between feature spaces (fields) of given types? Following Mackey, we show that such maps correspond one-to-one with convolutions using equivariant kernels, and characterize the space of such kernels.
1 Introduction
The paper introduces a unified formalism for G-CNNs by modelling feature maps as fields on homogeneous spaces and classifying them by symmetry group, base space, and field type. It further frames equivariant linear layers as convolutions with constrained equivariant kernels.
- Motivation: G-CNN approaches have proliferated across Euclidean space, spheres, and graphs, making their relationships and extension to new modalities difficult to assess.The paper motivates a systematic framework for comparing existing layers and designing equivariant networks for modalities such as diffusion tensor MRI.
- Feature spaces: Feature spaces are modelled as fields characterized by a symmetry group G, subgroup H defining B ≃ G/H, and an H-representation ρ specifying field type.Examples include scalar, vector, and tensor fields over homogeneous spaces such as the sphere.
- Scope and contribution: The paper presents a new formalism rather than claiming fundamentally new mathematics, connecting G-CNNs with fields and fiber bundles used in mathematics and physics.This formalism is intended to support systematic study and development of equivariant network architectures.
- Equivariant layers: Linear equivariant maps between feature spaces correspond one-to-one with convolutions using equivariant kernels.The kernels can be represented as matrix-valued functions on groups, coset spaces, or double coset spaces subject to linear constraints.
- Feature spaces: The theory uses fiber bundles to represent fields as sections whose fibers transform according to the representation ρ.For the sphere, SO(3) forms a principal SO(2) bundle over S2, while replacing SO(2) fibers with vector spaces yields associated bundles such as the tangent bundle.
- Transformation laws: The induced representation combines transformations of the homogeneous base space with transformations of the field fibers to define how fields transform.This representation provides the transformation law needed before studying equivariant maps between feature spaces.
2 Convolutional Feature Spaces
The paper models convolutional feature spaces as sections of associated vector bundles over homogeneous spaces, capturing both base-space symmetries and fiber transformation laws. This framework provides concrete global and local encodings of fields while accounting for non-trivial bundle topology.
- 2.1 Fiber Bundles: Fiber bundles parameterize isomorphic fibers over a base space and may have global topology distinct from a Cartesian product.The Möbius strip and the sphere’s tangent bundle illustrate non-trivial bundles that are locally product-like but globally distinct from corresponding products.
- 2.2 G as a Principal H-Bundle: A homogeneous space B can be represented as the coset space G/H, where H is the stabilizer subgroup of a chosen origin.The principal H-bundle has total space G, base space G/H, and fibers given by cosets, with right multiplication by H preserving each fiber.
- 2.3 The Associated Vector Bundle: The group action combines transformations of the homogeneous base with the representation-induced transformations of the fibers.A local frame generally acquires a fiber twist under the group action, described by an H-valued function relating transformed and reselected frames.
- 2.3 The Associated Vector Bundle: The associated vector bundle replaces each principal-bundle fiber H with a vector space V carrying an H-representation ρ.The quotient A = G ×_ρ V identifies (g,v) with (gh,ρ(h^-1)v), producing fibers whose feature vectors transform under changes of frame.
- 2.3 The Associated Vector Bundle: Feature maps are sections of the associated bundle, representing fields whose values remain in the fiber over each base point.Sections can be encoded globally by constrained functions f:G→V or locally by unconstrained functions on a trivializing neighbourhood U⊆G/H.
3 Equivariant Maps and Convolutions
The section proves that equivariant linear maps between feature spaces are convolution-like operations with constrained equivariant kernels. These kernels admit equivalent descriptions on G, G/H1, and H2\G/H1, while nonlinear layers require representation-dependent equivariance.
- Equivariant linear maps between induced representations can always be written as convolution-like integrals.
- Equivariance reduces a two-argument operator-valued kernel to a one-argument kernel depending on g^-1g′.
- The space of admissible kernels is in one-to-one correspondence with equivariant maps and can be characterized as bi-equivariant kernels on G.
- Equivariant kernels can also be represented on G/H1 or H2\G/H1, subject to the corresponding left, right, or double-coset constraints.
- For local fields, the construction becomes a ρ1-twisted cross-correlation on G/H1; semidirect products remove the ρ1 factor.
- Regular representations permit pointwise nonlinearities, whereas other representations require specially designed equivariant nonlinearities.
4 Implementation
Implementations depend on the symmetry group, homogeneous space, discretization, and field representation. Since kernel constraints are linear, implementations can construct constrained kernels from a basis, with sampling choices affecting practical behavior.
- Implementation details vary with the symmetry group G, homogeneous space G/H, discretization, and representation ρ.
- Equivariance constraints are linear, so equivariant kernels form a linear subspace and can be represented using a basis.
- On Euclidean grids, steerable kernel bases are pre-sampled, linearly combined during the forward pass, and applied with standard convolution routines.
- Euclidean sampling requires attention because it can introduce aliasing artifacts.
- Spherical CNNs may use spectral Fourier-space convolutions or spatial convolutions on sampled spherical grids.
5 Related Work
The paper situates its framework as a general treatment of fields and representations that organizes prior equivariant CNN work. It extends scalar-field treatments through associated bundles and induced representations while connecting to Mackey’s mathematics.
- Appendix D systematically classifies equivariant CNNs on homogeneous spaces using the paper’s framework.
- The framework generalizes closely related scalar-field theory to nontrivial fields using fiber bundles and induced representations.
- Prior work includes equivariant networks for finite groups, sets, graphs, and convolutional capsules.
- Mackey pioneered the study of induced representations and equivariant maps, with results essentially similar to this paper’s presented in a more abstract form.
6 Concrete Examples
The examples instantiate the theory for spherical CNNs and SE(3)-equivariant fields. Quotient spaces reduce kernel descriptions to radial or latitudinal domains, with representation choices determining the field behavior.
- The rotation group SO(3) and spherical CNNs: For SO(3) with H = SO(2), the homogeneous space G/H is the sphere S2, parameterized by spherical coordinates.
- The rotation group SO(3) and spherical CNNs: For spherical CNNs, the double coset space H\G/H is the segment [0, π), indexing latitudinal circle orbits.
- The rotation group SO(3) and spherical CNNs: Spherical equivariant kernels are matrix-valued functions on [0, π), mostly unconstrained except at the poles.
- The rotation group SO(3) and spherical CNNs: Trivial, regular, and standard SO(2) representations recover isotropic filters, non-isotropic spherical methods, and potential vector-field processing, respectively.
- SE(3): For SE(3) with H = SO(3), the homogeneous space G/H is R3 and the double coset space is radial.
- SE(3): SE(3)-equivariant maps use matrix-valued kernels on R3 satisfying SO(3)-equivariance under rotations.
7 Conclusion
The paper develops a general theory of equivariant convolutional networks on homogeneous spaces using fiber bundles and fields, connecting these networks to mathematical and physical theories.
- The theory uses fiber bundles and fields to describe equivariant convolutional networks on homogeneous spaces.
A General facts about Groups and Quotients
This section introduces groups, cosets, homogeneous spaces, sections, and related constructions used to represent geometric transformations and their fibers.
- Cosets partition a group, and their collection forms the quotient space G/H with canonical projection p(g) = gH.
- A transitive left action of G on G/H makes the quotient a homogeneous space, and every homogeneous space is isomorphic to some quotient G/H.
- A section chooses one representative s(x) from each coset, but the resulting constructions are independent of that choice.
- The function h(x, g) records the subgroup element needed to relate the chosen representative s(gx) to gs(x), because these representatives generally differ.
- Double cosets H2\G/H1 are the H2-orbits in G/H1, partitioning that quotient into disjoint subgroup orbits.
- For semidirect products, each group element uniquely decomposes as nh, allowing sections to be chosen in the normal subgroup N and simplifying h-related expressions.
- Haar measure provides an essentially unique invariant measure on locally compact groups, reducing to counting measure for discrete groups.
B Proofs
The proof framework imposes Mackey compatibility so that correlations and outputs satisfy the required field-space transformation laws.
- The result of correlation must lie in the output field space I2^G, so it must satisfy the Mackey condition.
- Because input functions satisfy the Mackey condition, right-H1 shifts of the kernel induce a corresponding symmetry in the correlation.
B.2 Kernels on H2\G/H1
The section establishes an equivalence between H2-equivariant kernels on G/H1 and a kernel space on the double-coset quotient.
- The space KC of H2-equivariant kernels on G/H1 is identified with an equivalent space KD associated with H2\G/H1.
- An isomorphism ΩK, constructed using a section γ of the double-coset space, maps between KD and KC.
- The proof verifies that the two kernel mappings are inverses.
- The construction decomposes y ∈ G/H1 as hγ(H2y)H1, although h need not be unique because H2 may not act freely on G/H1.
C Limitations of the Theory
The theory is restricted to homogeneous spaces and idealizes feature maps as fields over possibly continuous bases, leaving broader manifolds and discretized deep-network theory outside its scope.
- The theory covers fields over homogeneous spaces, not general or Riemannian manifolds.The restriction leverages the group action naturally available on homogeneous spaces.
- The framework models feature maps on possibly continuous base spaces, whereas implementations usually require discretization.The paper notes that analogous sampling-theorem or band-limit justifications, and a corresponding deep-network theory, remain undeveloped.
D Classification of Equivariant CNNs
Table 1 classifies G-CNN methods by their symmetry group, fiber subgroup, implied homogeneous base space, and field representation type.
- Methods are classified by the symmetry group G to which they are equivariant.
- The taxonomy records the subgroup H acting on fibers and the implied base space G/H.
- Methods are further distinguished by field type ρ, including regular, irreducible, and trivial representations.