Do ideas have shape? Idea registration as the continuous limit of artificial neural networks

Abstract

We introduce a GP generalization of ResNets (including ResNets as a particular case). We show that ResNets (and their GP generalization) converge, in the infinite depth limit, to a generalization of image registration variational algorithms. Whereas computational anatomy aligns images via warping of the material space, this generalization aligns ideas (or abstract shapes as in Plato's theory of forms) via the warping of the RKHS of functions mapping the input space to the output space. While the Hamiltonian interpretation of ResNets is not new, it was based on an Ansatz. We do not rely on this Ansatz and present the first rigorous proof of convergence of ResNets with trained weights and biases towards a Hamiltonian dynamics driven flow. Our constructive proof reveals several remarkable properties of ResNets and their GP generalization. ResNets regressors are kernel regressors with data-dependent warping kernels. Minimizers of $L_2$ regularized ResNets satisfy a discrete least action principle implying the near preservation of the norm of weights and biases across layers. The trained weights of ResNets with $L^2$ regularization can be identified by solving an autonomous Hamiltonian system. The trained ResNet parameters are unique up to the initial momentum whose representation is generally sparse. The kernel regularization strategy provides a provably robust alternative to Dropout for ANNs. We introduce a functional generalization of GPs leading to error estimates for ResNets. We identify the (EPDiff) mean fields limit of trained ResNet parameters. We show that the composition of warping regression blocks with reduced equivariant multichannel kernels (introduced here) recovers and generalizes CNNs to arbitrary spaces and groups of transformations.