Separation Power of Equivariant Neural Networks

Marco Pacini, Xiaowen Dong, Bruno Lepri, Gabriele Santin

Keywords: Geometric Deep Learning, Theory for Equivariant Neural Networks, Expressive Power

TL;DR: We present a new theoretical framework to investigate the separation power of equivariant neural networks.

Abstract:

The separation power of a machine learning model refers to its ability to distinguish between different inputs and is often used as a proxy for its expressivity. Indeed, knowing the separation power of a family of models is a necessary condition to obtain fine-grained universality results. In this paper, we analyze the separation power of equivariant neural networks, such as convolutional and permutation-invariant networks.
We first present a complete characterization of inputs indistinguishable by models derived by a given architecture. From this results, we derive how separability is influenced by hyperparameters and architectural choices—such as activation functions, depth, hidden layer width, and representation types. Notably, all non-polynomial activations, including ReLU and sigmoid, are equivalent in expressivity and reach maximum separation power. Depth improves separation power up to a threshold, after which further increases have no effect. Adding invariant features to hidden representations does not impact separation power. Finally, block decomposition of hidden representations affects separability, with minimal components forming a hierarchy in separation power that provides a straightforward method for comparing the separation power of models.

Primary Area: learning on graphs and other geometries & topologies

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Submission Number: 10616