Group equivariant convolutional neural networks (G-CNNs) have recently emerged as a very effective model class for learning from signals in the context of known symmetries. A wide variety of equivariant layers has been proposed for signals on 2D and 3D Euclidean space , graphs, and the sphere , and it has become difficult to see how all of these methods are related, and how they may be generalized. In this paper, we present a fairly general theory of equivariant convolutional networks . Convolutional feature spaces are described as fields over a homogeneous base space, such as the plane \mathbb{R . The theory enables a systematic classification of all existing G-CNNs in terms of their group of symmetry , base space, and field type (e. g. scalar, vector, or tensor field, etc.). In addition to this classification, we use Mackey theory to show that convolutions with equivariant kernels are the most general class of equivariant maps between such fields, thus establishing G-CNNs as a universal class of equivariant networks. The theory also explains how the space of equivariant kernelscan be parameterized for learning, thereby simplifying the development of G-CNNs for new spaces and symmetries. Finally, the theory introduces a rich geometric semantics to learned feature spaces, thus improving interpretability of deep networks, and establishing a connection to central ideas in mathematics and physics .
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 .
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Group equivariant convolutional neural networks
We present a general theory of Group equivariant Convolutional Neural Networks
R
have recently emerged as a very effective model class for learning from signals in the context of known symmetries. A wide variety of equivariant layers has been proposed for signals on 2D
on homogeneous spaces such as Euclidean space
D
3D Euclidean space , graphs, and
R
, and it has become difficult to see how all of these methods are related, and how they may be generalized. In this paper, we present a fairly general theory of equivariant convolutional networks . Convolutional feature spaces are described as fields over
. Feature maps in these networks represent fields on
R
such as the plane \mathbb{R
and layers are equivariant maps between spaces of fields
R
group of symmetry
symmetry group
R
(e. g. scalar, vector, or tensor field, etc.). In addition to this classification, we use Mackey theory to show that convolutions with equivariant kernels are
. We also consider a fundamental question: what is
R
class of equivariant maps between such fields, thus establishing G-CNNs as a universal class of equivariant networks. The theory also explains how the space of equivariant kernelscan be parameterized for learning, thereby simplifying the development of G-CNNs for new spaces and symmetries. Finally, the theory introduces a rich geometric semantics to learned feature spaces, thus improving interpretability of deep networks, and establishing a connection to central ideas in mathematics and physics
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