Covariant Compositional Networks For Learning Graphs

By Risi Kondor, H. Son, Horace Pan, Brandon M. Anderson, Shubhendu Trivedi. 2018

168 citations recorded by the source corpus.

Most existing neural networks for learning graphs address permutation invariance by conceiving of the network as a message passing scheme, where each node sums the feature vectors coming from its neighbors. We argue that this imposes a limitation on their representation power, and instead propose a new general architecture for representing objects consisting of a hierarchy of parts, which we call Covariant Compositional Networks (CCNs). Here, covariance means that the activation of each neuron must transform in a specific way under permutations, similarly to steerability in CNNs. We achieve covariance by making each activation transform according to a tensor representation of the permutation group, and derive the corresponding tensor aggregation rules that each neuron must implement. Experiments show that CCNs can outperform competing methods on standard graph learning benchmarks.

Subfields: Artificial Intelligence, Model Architectures, Graph Learning

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Citations in the corpus

E(n) Equivariant Graph Neural Networks (2021, background): “(Kondor et al., 2018) extends the equivariance of GNNs such that each neuron transforms in a speciﬁc way under permutations, but this extension only affects its permutation group and not translations or rotations in a geometric space.”
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…”
Understanding and mitigating gradient pathologies in physics-informed neural networks (2020, background): “Another example includes covariant neural networks [11], that are tailored to conform with the rotation and translation invariance present in many-body molecular systems.”
Learning Multimodal Graph-to-Graph Translation for Molecular Optimization (2018, background): “Previous work on graph encoders includes convolutional (Scarselli et al., 2009; Bruna et al., 2013; Henaff et al., 2015; Duvenaud et al., 2015; Niepert et al., 2016; Defferrard et al., 2016; Kondor et al., 2018) and recurrent architectures (Li et al., 2015; Dai et al., 2016; Lei et al., 2017).”
Set2Graph: Learning Graphs From Sets (2020, methodology): “…(Zaheer et al., 2017; Qi et al., 2017), graph learning (Kipf & Welling, 2016; Gilmer et al., 2017; 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.,…”
Tensor Field Networks: Rotation- and Translation-Equivariant Neural Networks for 3D Point Clouds (2018, background): “Kondor et al. [9] use tensor algebra to create neural network layers that extend Message Passing Neural Networks [10], but they are permutation group tensors (operating under permutation of the indices of the nodes), not geometric tensors.”
Provably Powerful Graph Networks (2019, background): “Kondor et al. (2018) identiﬁed several such linear and quadratic equivariant operators and showed that the resulting network can achieve excellent results on popular graph learning benchmarks.”
Gauge Equivariant Convolutional Networks and the Icosahedral CNN (2019, neutral): “…networks have been proposed for permutation-equivariant analysis and prediction of sets (Zaheer et al., 2017; Hartford et al., 2018), graphs (Kondor et al., 2018b; Hy et al., 2018; Maron et al., 2019), translations and rotations of the plane and 3D space (Oyallon & Mallat, 2015; Cohen &…”
N-body Networks: a Covariant Hierarchical Neural Network Architecture for Learning Atomic Potentials (2018, methodology): “Inspired by recent work on neural networks for representing graphs and other structured objects by covariant compositional neural architectures (Kondor et al., 2018), in this paper we take the idea of learnable multiscale representations one step further, and propose N –body networks, a neural…”
A General Theory of Equivariant CNNs on Homogeneous Spaces (2018, background): “Motivated by the success of CNNs, many researchers have worked on generalizations, leading to a growing body of work on Group equivariant CNNs (G-CNNs) for signals on Euclidean space and the sphere [1–7] as well as graphs [8, 9].”
3D G-CNNs for Pulmonary Nodule Detection (2018, background): “This is a burgeoning field of research, with G-CNNs being developed for discrete 2D rotation and reflection symmetries (15; 19; 16), continuous planar rotations (20; 21; 22), 3D rotations of spherical signals (23), and permutations of nodes in a graph (24).”
Universal Equivariant Multilayer Perceptrons (2020, background): “[21] Risi Kondor, Hy Truong Son, Horace Pan, Brandon Anderson, and Shubhendu Trivedi.”
On the Universality of Invariant Networks (2019, background): “Tasks involving graphs and hyper-graphs lead to symmetries defined by tensor products of permuta ons [2, 4].”
Message passing neural networks for molecular property prediction (2019, background): “Other approaches have explored the use of 3D atomic coordinates to augment the information provided to these models [46, 39, 25, 17, 18].”
Universal approximations of permutation invariant/equivariant functions by deep neural networks (2019, background): “Many researchers studied invariant/equivariant networks as [14], [23], [21], [4] and [17].”
GemNet: Universal Directional Graph Neural Networks for Molecules (2021, background): “Kondor et al. [37], Maron et al. [43], Morris et al. [45, 46] then investigated higher-order representations to circumvent this issue.”