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.

本文提出了一种称为深度函数机器（DFMs）的深度神经网络的概括，DFMs 作用于任意维度（可能是无限）的向量空间，并证明了一族 DFMs 对于输入数据的维度是不变的。使用这一概括，我们提出了一个新的理论，即有界非线性算子在函数空间之间的通用逼近。然后，我们建议 DFMs 提供了一种表达架构，用于以拓扑考虑为基础设计新的神经网络层类型，最后介绍了一种新的架构 RippLeNet，以实现具有分辨率不变性的计算机视觉，并在实践中取得了最先进的不变性。