Group Equivariant Convolutional Networks

By Taco Cohen, Max Welling. 2016

2,415 citations recorded by the source corpus.

We introduce Group equivariant Convolutional Neural Networks (G-CNNs), a natural generalization of convolutional neural networks that reduces sample complexity by exploiting symmetries. G-CNNs use G-convolutions, a new type of layer that enjoys a substantially higher degree of weight sharing than regular convolution layers. G-convolutions increase the expressive capacity of the network without increasing the number of parameters. Group convolution layers are easy to use and can be implemented with negligible computational overhead for discrete groups generated by translations, reflections and rotations. G-CNNs achieve state of the art results on CI- FAR10 and rotated MNIST.

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

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Citations in the corpus

Shortcut learning in deep neural networks (2020, background): “If the orientation of an object does not matter for its category, either dataaugmentation or hard-coded rotation invariance [123] can be applied.”
Group Normalization (2018, background): “However, in addition to orientations (SIFT [14], HOG [15], or [44,45]), there are many factors that could lead to grouping, e.”
Making Convolutional Networks Shift-Invariant Again (2019, background): “Recent work has strived to build in additional types of in-variances, such as rotation, reﬂection, and scaling (Sifre & Mallat, 2013; Bruna & Mallat, 2013; Esteves et al., 2018; Kanazawa et al., 2014; Worrall et al., 2017; Cohen & Welling, 2016).”
Rotation Equivariant CNNs for Digital Pathology (2018, methodology): “In this work, we focus on the straight-forward G-CNN method from [5] applied on discrete rotation/reﬂection groups.”
Perceiver: General Perception with Iterative Attention (2021, background): “…structures, such as videos (Tran et al., 2015) or audio (Ford et al., 2019), or for groups other than the group of linear translations (e.g. Cohen & Welling 2016; Bronstein et al. 2017; Esteves et al. 2018)); this however often requires a tremendous amount of researcher time and expertise.”
Global Pooling, More than Meets the Eye: Position Information is Encoded Channel-Wise in CNNs (2021, methodology): “To achieve invariant neural networks, a common strategy is to use equivariant operations on a per-layer basis [6], which then culminates in an invariant output.”
Why do deep convolutional networks generalize so poorly to small image transformations? (2018, methodology): “Alternatively, one could use specially designed features such as SIFT in which invariance is hard coded [7, 29, 30, 31] or neural network architectures that explicitly enforce invariance [36, 19, 5, 6, 11, 12, 47, 46, 8].”
E(n) Equivariant Graph Neural Networks (2021, background): “Group equivariant neural networks have demonstrated their effectiveness in a wide variety of tasks (Cohen & Welling, 2016; 2017; Weiler & Cesa, 2019; Rezende et al., 2019; Romero & Cordonnier, 2021).”
Directional Message Passing for Molecular Graphs (2020, background): “However, equivariant CNNs only achieve equivariance with respect to a discrete set of rotations (Cohen & Welling, 2016).”
Paying More Attention to Attention: Improving the Performance of Convolutional Neural Networks via Attention Transfer (2016, methodology): “This is similar to Group Equivariant CNN approach by Cohen & Welling (2016), however it is not a hard constraint.”
Invariant Representations without Adversarial Training (2018, background): “Recent work has provided a group-theoretic [4] analysis of the removal of covariate information (either by design or by augmentation), in which equivalences are drawn between finding invariant features and finding the quotient space of the domain over a covariate group action.”
Relational inductive biases, deep learning, and graph networks (2018, background): “Kondor et al. (2018) suggested that covariance 7 (Cohen and Welling, 2016; Kondor and Trivedi, 2018), rather than invariance to permutations of the nodes and edges is preferable, and proposed “covariant compositional networks” which can preserve structural information, and allow it to be ignored…”
Stacked Capsule Autoencoders (2019, methodology): “Geometrical Reasoning Other attempts at incorporating geometrical knowledge into neural networks include exploiting equivariance properties of group transformations (Cohen and Welling, 2016) or new types of convolutional ﬁlters (Oyallon and Mallat, 2015; Dieleman et al., 2016).”
Towards a Definition of Disentangled Representations (2018, background): “A long standing idea in ML is that such shortcomings can be reduced by introducing certain inductive biases into the model architecture that reﬂect the structure of the underlying data [12, 33, 21, 43].”
Covariant Compositional Networks For Learning Graphs (2018, background): “…an active ﬁeld of research, they differ from classical CNNs in a fundamental way: the internal feature representations in CNNs are equivariant to such transformations of the inputs as translation and rotations (Cohen & Welling, 2016a;b), the internal representations in MPNNs are fully invariant.”
Group Equivariant Stand-Alone Self-Attention For Vision (2020, methodology): “The seminal work by Cohen & Welling (2016) provides a recipe to extend the translation equivariance of CNNs to other symmetry groups to improve generalization and sample-efﬁciency further (see §2).”
Isometric Transformation Invariant and Equivariant Graph Convolutional Networks (2020, methodology): “Group equivariant convolutional neural networks were first proposed by Cohen & Welling (2016) for discrete groups.”
Generalizing Convolutional Neural Networks for Equivariance to Lie Groups on Arbitrary Continuous Data (2020, methodology): “One approach to constructing equivariant CNNs, ﬁrst introduced in Cohen and Welling (2016a), is to use standard convolutional kernels and transform them or the feature maps for each of the elements in the group.”
Set2Graph: Learning Graphs From Sets (2020, background): “…Veliˇckovi´c et al., 2017; Xu et al., 2019; Kondor et al., 2018; Maron et al., 2019b;a), learning images with rotational and reﬂectional symmetries (Cohen & Welling, 2016a;b; Dieleman et al., 2016; Worrall et al., 2017), learning functions on spheres (Cohen et al., 2018; Esteves et al., 2017)…”
General E(2)-Equivariant Steerable CNNs (2019, methodology): “As a result, we are able to implement a wide range of equivariant models, covering regular GCNNs [6, 7, 8, 9, 10, 11], classical Steerable CNNs [1], Harmonic Networks [12], gated Harmonic Networks [2], Vector Field Networks [13], Scattering Transforms [14, 15, 16, 17, 18] and entirely new…”