A General Theory of Equivariant CNNs on Homogeneous Spaces

By Taco Cohen, Mario Geiger, Maurice Weiler. 2018

370 citations recorded by the source corpus.

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.

Subfields: Artificial Intelligence, Computational Theory and Mathematics, Computer Vision and Pattern Recognition, Model Architectures

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Citations in the corpus

Group Equivariant Stand-Alone Self-Attention For Vision (2020, background): “…et al., 2018; Lenssen et al., 2018; Graham et al., 2020) to spherical rotations (Cohen et al., 2018; Worrall & Brostow, 2018; Weiler et al., 2018a; Cohen et al., 2019b), scaling (Marcos et al., 2018; Worrall & Welling, 2019; Sosnovik et al., 2020; Romero et al., 2020b) and more general symmetry…”
Generalizing Convolutional Neural Networks for Equivariance to Lie Groups on Arbitrary Continuous Data (2020, methodology): “For discrete groups this approach leads to exact equivariance and uses the so-called regular representation of the group (Cohen et al., 2019).”
General E(2)-Equivariant Steerable CNNs (2019, methodology): “We work in the framework of steerable CNNs [1, 2, 3, 4, 5] which provides a quite general theory for equivariant CNNs on homogeneous spaces, including Euclidean spaces R d as a speciﬁc instance.”
Gauge Equivariant Convolutional Networks and the Icosahedral CNN (2019, positive): “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 Group-Theoretic Framework for Data Augmentation (2019, background): “Starting from [30, 45, 61, 14], a line of research focuses on designing (or learning) neural net architectures that are invariant to certain symmetries [31, 27, 70, 20, 22, 19], equivariant to SO(3) group [21, 28, 29], and others [58, 42, 69, 41].”
Universal Equivariant Multilayer Perceptrons (2020, background): “[5] Taco S Cohen, Mario Geiger, and Maurice Weiler.”
B-Spline CNNs on Lie Groups (2019, methodology): “In other words, given the definition of G cross-correlation in (6), denoted with ?, and gauge correlation in (18) or (Cohen et al., 2019, Eq.”
Attentive Group Equivariant Convolutional Networks (2020, methodology): “2): Resultantly, the computational cost of a group convolution is roughly equivalent to that of a spatial convolution with a ﬁlter bank of size N ˜ c × | H | (Cohen & Welling, 2016a; Worrall & Welling, 2019; Cohen et al., 2019b).”
On Learning Sets of Symmetric Elements (2020, background): “From a theoretical point of view, several papers studied the properties of equivariant layers (Ravanbakhsh et al., 2017; Kondor & Trivedi, 2018; Cohen et al., 2019a) and characterized the expressive power of models that use such layers (Yarotsky, 2018; Maron et al., 2019c; Keriven & Peyr´e, 2019;…”
Probabilistic Symmetries and Invariant Neural Networks (2020, background): “Recent work shows that they are the only equivariant linear maps (Kondor and Trivedi, 2018; Cohen et al., 2019).”
Wavelet Networks: Scale Equivariant Learning From Raw Waveforms (2020, methodology): “It is well known, however, that equivariance to an arbitrary group can only be obtained via group convolutions (or reparametrizations thereof) defined in the corresponding group [31, 14, 3].”
GemNet: Universal Directional Graph Neural Networks for Molecules (2021, background): “Equivariance and invariance have recently emerged as one of the foundational principles of modern neural networks [13, 15].”