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Intertwiners between Induced Representations (with Applications to the Theory of Equivariant Neural Networks)
Taco S. Cohen, Mario Geiger, Maurice Weiler
TL;DR
Equivariant G-CNN layers must intertwine induced representations, but the general structure of such intertwiners requires a unified framework. The paper develops that framework and shows that equivariant maps between induced representations are precisely representable by twisted convolutions or cross-correlations, with kernels reducible to double-coset spaces. This generalizes the regular-representation result and supports systematic construction of equivariant filters.
Problem
Equivariant networks need a general characterization of intertwiners between induced representations for feature spaces containing scalar, vector, or tensor fields.
Source: §1 · pp. 2–3, §4 · p. 11Method
The paper uses elementary calculations and equivalent realizations of induced representations to analyze equivariant maps and their kernels on groups, homogeneous spaces, and double-coset spaces.
Source: §1.1 · p. 4Results
Any equivariant map between induced representations can be written as a twisted convolution or cross-correlation, with equivariant kernels equivalent to kernels on H2\G/H1.
Source: §1 · p. 3, §1.1 · p. 4Takeaways & Limitations
The framework provides a systematic route to constructing equivariant G-CNN layers and computing bases for equivariant filters across groups and manifolds.
Source: §1.1 · p. 4The function realization technically assumes a continuous section of G/H into G defined almost everywhere, while the construction also involves potentially nonunique subgroup decompositions.
Source: §3.2 · p. 9, §4.3 · p. 15
Abstract
from arXiv · showhide
Group equivariant and steerable convolutional neural networks (regular and steerable G-CNNs) have recently emerged as a very effective model class for learning from signal data such as 2D and 3D images, video, and other data where symmetries are present. In geometrical terms, regular G-CNNs represent data in terms of scalar fields ("feature channels"), whereas the steerable G-CNN can also use vector or tensor fields ("capsules") to represent data. In algebraic terms, the feature spaces in regular G-CNNs transform according to a regular representation of the group G, whereas the feature spaces in Steerable G-CNNs transform according to the more general induced representations of G. In order to make the network equivariant, each layer in a G-CNN is required to intertwine between the induced representations associated with its input and output space. In this paper we present a general mathematical framework for G-CNNs on homogeneous spaces like Euclidean space or the sphere. We show, using elementary methods, that the layers of an equivariant network are convolutional if and only if the input and output feature spaces transform according to an induced representation. This result, which follows from G.W. Mackey's abstract theory on induced representations, establishes G-CNNs as a universal class of equivariant network architectures, and generalizes the important recent work of Kondor & Trivedi on the intertwiners between regular representations.
1 Introduction
The paper develops induced G-CNNs, extending regular scalar-field networks to vector and tensor fields on homogeneous spaces. It proves that equivariant maps between induced representations are convolutional and characterizes kernels in forms useful for implementation.
1 Introduction
- Induced G-CNNs extend regular scalar-field networks to vector and tensor fields, with feature spaces transforming under induced representations.Regular G-CNNs use scalar fields on groups or homogeneous spaces, whereas steerable G-CNNs use more general fields associated with induced representations.Source: §1 · p. 2
- Any equivariant map between induced representations can be written as a twisted convolution or cross-correlation, generalizing the regular-representation result of Kondor and Trivedi [2018].The paper studies intertwiners between representations induced from potentially different subgroups and identifies their convolutional form.Source: §1 · p. 3, §1.1 · p. 4
- The induced transformation law moves each field value to its new position and transforms the vector itself through a subgroup representation.For vector fields, the subgroup action rotates the vector in its local tangent plane; higher-order tensors use different subgroup representations.Source: §1 · pp. 2–3, Figure 1 · p. 2
1.1 Outline and Summary of Results
- A lifting isomorphism establishes equivalence between geometrically natural vector-bundle sections and symmetric vector-valued functions on G.The section realization is efficient for computer storage, while the function-on-G realization makes formulas less dependent on an arbitrary frame.Source: §1.1 · p. 4
- The kernel spaces in the group and homogeneous-space realizations are equivalent to kernels on the double-coset space H2\G/H1.Moving to the double-coset representation largely resolves the equivariance constraints and simplifies construction of equivariant kernels.Source: §1.1 · p. 4
- The framework provides a route to computing bases for equivariant filters and is applied to G = SO(3) and G = SE(3).The paper connects its concrete derivations to Mackey’s theory before computing bases for KC in these two cases.Source: §1.1 · p. 4
2 Mathematical Background
The mathematical background introduces quotient and homogeneous spaces, sections, double cosets, semidirect products, and Haar measure. These constructions organize group actions and the integrations used later for induced representations and equivariant kernels.
2.1 General facts about Groups and Quotients
- A quotient space G/H consists of left cosets gH that partition G, with G acting transitively on the resulting homogeneous space.The canonical projection maps each group element to its coset, while the induced action on G/H is well defined and associative.Source: §2.1 · p. 5
- A section s:G/H→G chooses one representative from each coset, and the constructions are independent of which section is chosen.The paper assumes, when convenient, that the identity coset maps to the identity element.Source: §2.1 · p. 5
- The h-function records the subgroup correction relating gs(x) to the chosen representative s(gx), since these elements generally differ.This correction is central to defining induced representations and is illustrated through the D3 Cayley diagram.Source: §2.1 · p. 5, Figure 2 · p. 6
2.2 Double cosets
- The double-coset space H2\G/H1 is the space of H2-orbits in G/H1, grouping points that lie in the same subgroup orbit.Because H2 need not act transitively or freely on G/H1, stabilizers can occur within these orbits.Source: §2.2 · p. 7
2.3 Semidirect products
- For a semidirect product G=N⋊H, each element decomposes uniquely as nh, and left H-cosets depend only on the N component.A section can therefore be chosen to output elements of N, simplifying expressions involving the h-function.Source: §2.3 · pp. 7–8
2.4 Haar measure
- Haar measure is an essentially unique invariant measure for integration over locally compact groups, reducing to counting measure for discrete groups.Integration over G/H can be defined through an integral over G.Source: §2.4 · p. 8
3 Induced Representations
The paper defines induced representations in two equivalent realizations: Mackey functions on G and vector-valued functions on G/H. The quotient-space realization is geometrically natural and implementation-friendly, while a lifting isomorphism establishes equivalence between the two forms.
3 Induced Representations
- For implementations, the G/H realization is preferred because it stores geometric fields efficiently, although choosing a section makes its equations more cumbersome.The Mackey-function realization is easier to handle mathematically, whereas the quotient realization more clearly conveys geometric meaning.Source: §3 · p. 9
3.1 Realization through Mackey Functions
- Mackey functions are functions on G satisfying the subgroup symmetry f(gh)=ρ(h^-1)f(g), with G acting by left translation.This realization is mathematically convenient for analyzing induced representations.Source: §3.1 · p. 9, §3 · p. 9
3.2 Realization through Functions on G/H
- Induced G-CNN feature spaces can be realized as vector-valued functions on G/H, with group actions transforming positions and fiber vectors through the subgroup representation.The transformation moves f(u^-1x) to position x and applies the representation ρ of H to the vector.Source: §3.2 · p. 9
3.3 Equivalence of the Realizations
- The lifting map between quotient-space functions and Mackey functions is bijective and commutes with the induced G-actions, so the two realizations are isomorphic.Mackey functions are determined by their values on coset representatives and can be reconstructed using the subgroup factor h(g).Source: §3.3 · p. 10
3.4 Some basic properties of induction
- The section also introduces standard induction properties, including induction in stages and preservation of direct sums under induction.These facts are stated as basic representation-theoretic results used in the paper’s framework.Source: §3.4 · p. 10
4 Intertwiners: Elementary Approach
The elementary approach characterizes equivariant maps between induced representations as convolutions or cross-correlations with constrained equivariant kernels. These kernel constraints can be reduced from functions on G to simpler kernels on quotient and double-coset spaces.
4 Intertwiners: Elementary Approach
- Every intertwiner between induced representations can be expressed as a convolution or cross-correlation with an equivariant kernel.The result applies in both Mackey-function and quotient-space realizations.Source: §4 · p. 11, §4.2 · p. 14
4.1.1 Equivariance ⇔Convolution
- For the Mackey-function realization, equivariance reduces a two-argument kernel to a one-argument cross-correlation kernel satisfying bi-equivariance constraints.The admissible kernels are in one-to-one correspondence with the bi-equivariant kernel space K_G.Source: §4.1 · p. 11, §4.1.1 · p. 11, §4.1.4 · p. 12
4.1.4 Resolving the right-equivariance constraint
- Resolving the right H1-equivariance identifies bi-equivariant kernels on G with left-equivariant kernels on the quotient G/H1.The quotient kernels take values in Hom(V1,V2) and retain the remaining H2-equivariance condition.Source: §4.1.4 · pp. 12–13
4.2 Intertwiners for IC
- For the quotient-space realization, intertwiners are twisted cross-correlations using kernels on G/H1 that belong to the same constrained space K_C.For semidirect product groups, the ρ1-twist disappears.Source: §4.2 · p. 14
4.3 Resolving the left-equivariance constraint
- The remaining left-equivariance constraint can be resolved by passing from K_C to a kernel space K_D indexed by the double-coset structure.The construction uses a section of H2\G/H1, although the relevant subgroup action need not be free.Source: §4.3 · pp. 14–15
5 Intertwiners: Abstract Approach
The abstract approach combines Frobenius reciprocity with Mackey’s lemma to analyze intertwiners between induced representations. Frobenius reciprocity converts the problem into equivariant maps involving a restricted representation, which Mackey’s lemma then decomposes.
5 Intertwiners: Abstract Approach
- Frobenius reciprocity identifies G-equivariant maps into an induced representation with H-equivariant maps from the restricted source representation.It expresses induction as the adjoint of restriction.Source: §5 · p. 16
- Applying reciprocity to two induced representations reduces their intertwiner space to H2-equivariant maps from the restriction of the first induced representation to the second subgroup representation.Mackey’s lemma supplies the needed decomposition of that restricted induced representation.Source: §5 · p. 16
H2 IndG
Mackey’s lemma decomposes the restricted induced representation using double cosets and stabilizer subgroups, yielding a complete decomposition of the intertwiner space. When the stabilizer is trivial almost everywhere, the equivariance constraint simplifies accordingly.
H2 IndG
- Mackey’s lemma decomposes the restriction of an induced representation over the double-coset space H2\G/H1.Each double-coset component is associated with a stabilizer subgroup and a corresponding representation.Source: H2 IndG · p. 16
- Mackey’s formula for invariants gives a complete decomposition of the intertwiners between two induced representations.The decomposition is formulated using the double-coset space and stabilizer representations.Source: H2 IndG · p. 16
- Understanding the intertwiner space therefore reduces to analyzing the double-coset space and the associated stabilizer subgroup.The stabilizer constraint is trivial almost everywhere when the stabilizer subgroup is trivial.Source: H2 IndG · p. 16
6 Examples
The examples specialize the general intertwiner framework to spherical SO(3) layers and SE(3) layers, reducing equivariance to constrained kernels on double-coset spaces.
6.1 Intertwiners for SO(3)
- For SO(3) fields on the sphere, G/H is S2, while the double-coset space H\G/H is the interval [0, π] parameterized by β.The double cosets correspond to circles of constant latitude around the Z-axis.Source: §6.1 · p. 17
- The SO(3) stabilizer is SO(2) at the north and south poles but trivial elsewhere, so Mackey’s constraint is almost everywhere vacuous.This makes the kernel constraint unconstrained away from the poles.Source: §6.1 · p. 17
H V1, IndG
- Consequently, equivariant spherical kernels are matrix-valued functions from [0, π] to R^{n×m}, matching zonal scalar spherical convolution.The kernel depends on the double-coset coordinate β rather than the azimuthal coordinate α.Source: H V1, IndG · pp. 17–18
- The SO(3) kernel basis can be computed from the linear equivariance constraint, or equivalently parameterized by a reduced kernel β ↦ κ̄(β).The reduced representation maps through the isomorphism ΩK from double-coset kernels to coset kernels.Source: H V1, IndG · p. 18
6.2 Intertwiners for SE(3)
- For SE(3), double cosets are radii R ∈ [0, ∞), each with stabilizer SO(2)_z, yielding a constrained kernel that can be solved analytically or numerically.The resulting layer is a 3D convolution with a special constrained kernel.Source: §6.2 · pp. 18–19
- The coset-to-kernel map for SE(3) reconstructs the spatial kernel by rotating the double-coset kernel according to the chosen group element h.This realizes the constrained double-coset kernel as the kernel used in the 3D convolution.Source: §6.2 · pp. 18–19
7 Conclusion
The paper characterizes layers of generalized steerable G-CNNs as intertwiners between induced representations and expresses them through constrained correlation kernels.
7 Conclusion
- Intertwiners between induced representations are characterized generically as constrained correlation kernels on coset or double-coset spaces.The framework applies across a wide range of groups and homogeneous spaces.Source: §7 · p. 19
- This characterization makes constructing steerable G-CNNs for new groups and spaces a relatively straightforward calculation while yielding the most general intertwiner parameterization.The conclusion identifies this as the practical consequence of the framework.Source: §7 · p. 19