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Recent advances in adversarial attacks and Wasserstein GANs have advocatedfor use of neural networks with restricted Lipschitz constants. Motivated bythese observations, we study the recently introduced GroupSort neural networks,with constraints on the weights, and make a theoretical step towards a betterunderstanding of their expressive power. We show in particular how thesenetworks can represent any Lipschitz continuous piecewise linear functions. Wealso prove that they are well-suited for approximating Lipschitz continuousfunctions and exhibit upper bounds on both the depth and size. To conclude, theefficiency of GroupSort networks compared with more standard ReLU networks isillustrated in a set of synthetic experiments.

Approximating Lipschitz continuous functions with GroupSort neural networks