Group Equivariant Conditional Neural Processes

We present the group equivariant conditional neural process (EquivCNP), a metalearning method with permutation invariance in a data set as in conventional conditional neural processes (CNPs), and it also has transformation equivariance in data space. Incorporating group equivariance, such as rotation and scaling equivariance, provides a way to consider the symmetry of real-world data. We give a decomposition theorem for permutation-invariant and group-equivariant maps, which leads us to construct EquivCNPs with an infinite-dimensional latent space to handle group symmetries. In this paper, we build architecture using Lie group convolutional layers for practical implementation. We show that EquivCNP with translation equivariance achieves comparable performance to conventional CNPs in a 1D regression task. Moreover, we demonstrate that incorporating an appropriate Lie group equivariance, EquivCNP is capable of zero-shot generalization for an image-completion task by selecting an appropriate Lie group equivariance. Data symmetry has played a significant role in the deep neural networks. In particular, a convolutional neural network, which play an important part in the recent achievements of deep neural networks, has translation equivariance that preserves the symmetry of the translation group. From the same point of view, many studies have aimed to incorporate various group symmetries into neural networks, especially convolutional operation (Cohen et al., 2019; Defferrard et al., 2019; Finzi et al., 2020).