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Parameter-space and function-space provide two different duality frames inwhich to study neural networks. We demonstrate that symmetries of networkdensities may be determined via dual computations of network correlationfunctions, even when the density is unknown and the network is not equivariant.Symmetry-via-duality relies on invariance properties of the correlationfunctions, which stem from the choice of network parameter distributions. Inputand output symmetries of neural network densities are determined, which recoverknown Gaussian process results in the infinite width limit. The mechanism mayalso be utilized to determine symmetries during training, when parameters arecorrelated, as well as symmetries of the Neural Tangent Kernel. We demonstratethat the amount of symmetry in the initialization density affects the accuracyof networks trained on Fashion-MNIST, and that symmetry breaking helps onlywhen it is in the direction of ground truth.

Symmetry-via-Duality: Invariant Neural Network Densities from Parameter-Space Correlators