Separation Power of Equivariant Neural Networks

The separation power of a machine learning model refers to its capacity to distinguish distinct inputs, and it is often employed as a proxy for its expressivity. In this paper, we propose a theoretical framework to investigate the separation power of equivariant neural networks with point-wise activations. Using the proposed framework, we can derive an explicit description of inputs indistinguishable by a family of neural networks with given architecture, demonstrating that it remains unaffected by the choice of non-polynomial activation function employed. We are able to understand the role played by activation functions in separability. Indeed, we show that all non-polynomial activations, such as ReLU and sigmoid, are equivalent in terms of expressivity, and that they reach maximum discrimination capacity. We demonstrate how assessing the separation power of an equivariant neural network can be simplified to evaluating the separation power of minimal representations. We conclude by illustrating how these minimal components form a hierarchy in separation power.

A Characterization Theorem for Equivariant Networks with Point-wise Activations

Equivariant neural networks have shown improved performance, expressiveness and sample complexity on symmetrical domains. But for some specific symmetries, representations, and choice of coordinates, the most common point-wise activations, such as ReLU, are not equivariant, hence they cannot be employed in the design of equivariant neural networks. The theorem we present in this paper describes all possible combinations of finite-dimensional representations, choice of coordinates and point-wise activations to obtain an exactly equivariant layer, generalizing and strengthening existing characterizations. Notable cases of practical relevance are discussed as corollaries. Indeed, we prove that rotation-equivariant networks can only be invariant, as it happens for any network which is equivariant with respect to connected compact groups. Then, we discuss implications of our findings when applied to important instances of exactly equivariant networks. First, we completely characterize permutation equivariant networks such as Invariant Graph Networks with point-wise nonlinearities and their geometric counterparts, highlighting a plethora of models whose expressive power and performance are still unknown. Second, we show that feature spaces of disentangled steerable convolutional neural networks are trivial representations.