A tradeoff between universality of equivariant models and learnability of symmetries

TL;DR

This paper establishes an impossibility result showing the tradeoff between the universality of equivariant models and their ability to learn symmetries, with implications for neural network design and symmetry approximation.

Contribution

It formalizes the limitations of learning symmetries with equivariant functions and analyzes neural network classes, extending the convolution theorem to non-homogeneous spaces.

Findings

Impossibility of simultaneously learning symmetries and equivariant functions under certain conditions

Theoretical analysis of linearly equivariant networks as a useful model

Generalization of the convolution theorem to non-homogeneous spaces

Abstract

We prove an impossibility result, which in the context of function learning says the following: under certain conditions, it is impossible to simultaneously learn symmetries and functions equivariant under them using an ansatz consisting of equivariant functions. To formalize this statement, we carefully study notions of approximation for groups and semigroups. We analyze certain families of neural networks for whether they satisfy the conditions of the impossibility result: what we call ``linearly equivariant'' networks, and group-convolutional networks. A lot can be said precisely about linearly equivariant networks, making them theoretically useful. On the practical side, our analysis of group-convolutional neural networks allows us generalize the well-known ``convolution is all you need'' theorem to non-homogeneous spaces. We additionally find an important difference between group convolution and semigroup convolution.