  In this paper we propose a generalization of deep neural networks called deep
function machines (DFMs). DFMs act on vector spaces of arbitrary (possibly
infinite) dimension and we show that a family of DFMs are invariant to the
dimension of input data; that is, the parameterization of the model does not
directly hinge on the quality of the input (eg. high resolution images). Using
this generalization we provide a new theory of universal approximation of
bounded non-linear operators between function spaces. We then suggest that DFMs
provide an expressive framework for designing new neural network layer types
with topological considerations in mind. Finally, we introduce a novel
architecture, RippLeNet, for resolution invariant computer vision, which
empirically achieves state of the art invariance.
