Intertwiners between Induced Representations (with Applications to the Theory of Equivariant Neural Networks)

By Taco Cohen, Mario Geiger, Maurice Weiler. 2018

49 citations recorded by the source corpus.

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.

Subfields: Artificial Intelligence, Materials Chemistry, Neurology

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Citations in the corpus

General E(2)-Equivariant Steerable CNNs (2019, methodology): “Since the groups ( R 2 , +) (cid:111) G are semidirect products, one can uniquely decompose any of their elements into a product tg where t ∈ ( R 2 , +) and g ∈ G [3] which we will do in the rest of the paper.”
Tensor Field Networks: Rotation- and Translation-Equivariant Neural Networks for 3D Point Clouds (2018, background): “Further discussion of related theory can be found in [20], [21], and [22].”
Gauge Equivariant Convolutional Networks and the Icosahedral CNN (2019, methodology): “This approach works for any homogeneous space , where by deﬁnition it is possible to move from any point p ∈ M to any other point q ∈ M using an appropriate symmetry transformation (Kondor & Trivedi, 2018; Cohen et al., 2018c;a).”
A General Theory of Equivariant CNNs on Homogeneous Spaces (2018, background): “A preliminary version of this paper appeared as [32].”
3D G-CNNs for Pulmonary Nodule Detection (2018, background): “Although the general theory of G-CNNs is now well established [15, 16, 17, 18], a lot of work remains in developing easy to use group convolution layers for various kinds of input data with various kinds of symmetries.”
Explorations in Homeomorphic Variational Auto-Encoding (2018, background): “Equivariant networks, investigated in Cohen & Welling (2016; 2017); Worrall et al. (2017); Weiler et al. (2018); Cohen et al. (2018b) reduce the complexity of a learning task by taking a quotient over group orbits which explain a subset of dimensions of the data manifold.”
Group Equivariant Capsule Networks (2018, methodology): “…more general set of methods are the group convolutional networks [Cohen and Welling, 2016, Dieleman et al., 2016] and its applications like Spherical CNNs in SO (3) [Cohen et al., 2018] and Steerable CNNs in SO (2) [Cohen and Welling, 2017], which both result in special convolution realizations.”
Small Data Challenges in Big Data Era: A Survey of Recent Progress on Unsupervised and Semi-Supervised Methods (2019, background): “Before the concept of learning Transformation-Equivariance Representations (TER) was proposed by Hinton et al. [5], [60], [63], [64], most attentions have been paid on the transformation invariance criterion to train supervised models by minimizing the classiﬁcation errors on labeled images…”