What Kinds of Functions do Deep Neural Networks Learn? Insights from Variational Spline Theory

We develop a variational framework to understand the properties of functions learned by deep neural networks with ReLU activation functions fit to data. We propose a new function space, which is reminiscent of classical bounded variation spaces, that captures the compositional structure associated with deep neural networks. We derive a representer theorem showing that deep ReLU networks are solutions to regularized data fitting problems in this function space. The function space consists of compositions of functions from the (non-reflexive) Banach spaces of second-order bounded variation in the Radon domain. These are Banach spaces with sparsity-promoting norms, giving insight into the role of sparsity in deep neural networks. The neural network solutions have skip connections and rank bounded weight matrices, providing new theoretical support for these common architectural choices. The variational problem we study can be recast as a finite-dimensional neural network training problem with regularization schemes related to the notions of weight decay and path-norm regularization. Finally, our analysis builds on techniques from variational spline theory, providing new connections between deep neural networks and splines.