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Motivated by classical work on the numerical integration of ordinarydifferential equations we present a ResNet-styled neural network architecturethat encodes non-expansive (1-Lipschitz) operators, as long as the spectralnorms of the weights are appropriately constrained. This is to be contrastedwith the ordinary ResNet architecture which, even if the spectral norms of theweights are constrained, has a Lipschitz constant that, in the worst case,grows exponentially with the depth of the network. Further analysis of theproposed architecture shows that the spectral norms of the weights can befurther constrained to ensure that the network is an averaged operator, makingit a natural candidate for a learned denoiser in Plug-and-Play algorithms.Using a novel adaptive way of enforcing the spectral norm constraints, we showthat, even with these constraints, it is possible to train performant networks.The proposed architecture is applied to the problem of adversarially robustimage classification, to image denoising, and finally to the inverse problem ofdeblurring.

Designing Stable Neural Networks using Convex Analysis and ODEs