Learning Partial Equivariances from Data

Monte Carlo Approximation of Group Convolutions

We can approximate Group Convolutions on the expectation by uniformly sampling group actions $v_j$.

$$(\psi \hat{\ast }f)(u_i)=\sum _j\psi (v_j^{-1}{u}_i)f(v_j){\bar{\mu }}_{\mathcal{G}}(v_j)$$

Main idea

1. Prioritize sampling of specific group elements during the group convolution by learning a probability distribution over them.
2. 1D continuous groups: use reparametrization trick on the Lie algebra of the group, which is uniform over a connected set of group elements but zero otherwise. $\to$ Partial Equivariance
3. 1D discrete groups: Bernoulli Distribution over all possible element combinations

Citations:

• Self-Supervised Detection of Perfect and Partial Input-Dependent Symmetries
• Color Equivariant Convolutional Networks
• Equivariance-aware architectural optimization of neural networks
• Approximation-Generalization Trade-offs under (Approximate) Group Equivariance