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：本文提出了一种深度学习中的假设空间，该空间采用深度神经网络（DNNs）。通过将DNN视为两个变量（物理变量和参数变量）的函数，我们考虑了参数变量位于由DNN的预设深度和宽度确定的权重矩阵和偏置集合中时的DNN原始集。随后，我们在弱*拓扑下完成对原始DNN集的线性张成，从而构造出一个关于物理变量的Banach函数空间。我们证明如此构造的Banach空间是一个再生核Banach空间（RKBS），并构造了其再生核。我们研究了该RKBS中的两种学习模型——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来展开。这些表示定理揭示出，这些学习模型的解可以表示为有限个由给定数据和再生核确定的核函数片的线性组合。