  We develop a variational framework to understand the properties of the
functions learned by neural networks fit to data. We propose and study a family
of continuous-domain linear inverse problems with total variation-like
regularization in the Radon domain subject to data fitting constraints. We
derive a representer theorem showing that finite-width, single-hidden layer
neural networks are solutions to these inverse problems. We draw on many
techniques from variational spline theory and so we propose the notion of
polynomial ridge splines, which correspond to single-hidden layer neural
networks with truncated power functions as the activation function. The
representer theorem is reminiscent of the classical reproducing kernel Hilbert
space representer theorem, but we show that the neural network problem is posed
over a non-Hilbertian Banach space. While the learning problems are posed in
the continuous-domain, similar to kernel methods, the problems can be recast as
finite-dimensional neural network training problems. These neural network
training problems have regularizers which are related to the well-known weight
decay and path-norm regularizers. Thus, our result gives insight into
functional characteristics of trained neural networks and also into the design
neural network regularizers. We also show that these regularizers promote
neural network solutions with desirable generalization properties.
