This paper introduces a hypothesis space for deep learning that employs deep neural networks (DNNs). By treating a DNN as a function of two variables, the physical variable and parameter variable, we consider the primitive set of the DNNs for the parameter variable located in a set of the weight matrices and biases determined by a prescribed depth and widths of the DNNs. We then complete the linear span of the primitive DNN set in a weak* topology to construct a Banach space of functions of the physical variable. We prove that the Banach space so constructed is a reproducing kernel Banach space (RKBS) and construct its reproducing kernel. We investigate two learning models, regularized learning and minimum interpolation problem in the resulting RKBS, by establishing representer theorems for solutions of the learning models. The representer theorems unfold that solutions of these learning models can be expressed as linear combination of a finite number of kernel sessions determined by given data and the reproducing kernel.

翻译：本文提出了一个基于深度神经网络（DNN）的深度学习假设空间。通过将DNN视为两个变量（物理变量与参数变量）的函数，我们考虑参数变量位于由预设深度和宽度确定的权重矩阵与偏置集合中的原始DNN集。随后，我们在弱*拓扑下完成原始DNN集的线性张成，构建出关于物理变量的函数Banach空间。我们证明了所构建的Banach空间是再生核Banach空间（RKBS），并构造了其再生核。在该RKBS中研究了两种学习模型——正则化学习与最小插值问题，通过建立学习模型解的表示定理。表示定理揭示出这些学习模型的解可表示为有限个由给定数据与再生核确定的核会话的线性组合。