Symmetric functions, which take as input an unordered, fixed-size s et, find practical application in myriad physical settings based on indistinguishable points or particles, and are also used as intermediate building blocks to construct networks with other invariances. Symmetric functions
are known to be universally representable by neural networks that enforce permutation invariance. However the theoretical tools that characterize the approximation, optimization and generalization of typical networks fail to adequately characterize architectures that enforce invariance.