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arXiv · 1901.09342

On the Universality of Invariant Networks

Haggai Maron·Ethan Fetaya·Nimrod Segol·Yaron Lipman
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Abstract

Constraining linear layers in neural networks to respect symmetry transformations from a group $G$ is a common design principle for invariant networks that has found many applications in machine learning. In this paper, we consider a fundamental question that has received little attention to date: Can these networks approximate any (continuous) invariant function? We tackle the rather general case where $G\leq S_n$ (an arbitrary subgroup of the symmetric group) that acts on $\mathbb{R}^n$ by permuting coordinates. This setting includes several recent popular invariant networks. We present two main results: First, $G$-invariant networks are universal if high-order tensors are allowed. Second, there are groups $G$ for which higher-order tensors are unavoidable for obtaining universality. $G$-invariant networks consisting of only first-order tensors are of special interest due to their practical value. We conclude the paper by proving a necessary condition for the universality of $G$-invariant networks that incorporate only first-order tensors.

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Haggai Maron, Ethan Fetaya, Nimrod Segol, Yaron Lipman. 2019-01-27. On the Universality of Invariant Networks. https://arxiv.org/abs/1901.09342

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