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 ≤ S n (an arbitrary subgroup of the symmetric group) that acts on 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 firstorder 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 firstorder tensors. Lastly, we propose a conjecture stating that this condition is also sufficient.
REF demonstrated that higher-order tensors are sufficient and necessary for universal invariant networks.
On the Universality of Invariant Networks
ArXiv
ArXiv
Mathematics Computer Science