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.

该论文介绍了一种用于深度学习的假设空间，利用深度神经网络（DNNs），通过将 DNN 视为两个变量的函数，即物理变量和参数变量，并考虑 DNN 的原始集合，这些集合位于由 DNN 的深度和宽度确定的权重矩阵和偏差的集合，然后通过在弱 * 拓扑中完成原始 DNN 集合的线性张量构建物理变量的函数的 Banach 空间，我们证明了所构建的 Banach 空间是一个再生内核 Banach 空间（RKBS），并构建了其再生内核。我们通过建立学习模型的 representer 定理，研究了结果 RKBS 中的正则化学习和最小插值问题，representer 定理表明这些学习模型的解可以表示为给定数据和再生内核确定的有限数量的内核会话的线性组合。

深度学习的假设空间

Hypothesis Spaces for Deep Learning

Graph convolutional neural network (GCNN) operates on graph domain and it has
achieved a superior performance to accomplish a wide range of tasks. In this
paper, we introduce a Barron space of functions on a compact domain of graph
signals. We prove that the proposed Barron space is a reproducing kernel Banach
space, it can be decomposed into the union of a family of reproducing kernel
Hilbert spaces with neuron kernels, and it could be dense in the space of
continuous functions on the domain. Approximation property is one of the main
principles to design neural networks. In this paper, we show that outputs of
GCNNs are contained in the Barron space and functions in the Barron space can
be well approximated by outputs of some GCNNs in the integrated square and
uniform measurements. We also estimate the Rademacher complexity of functions
with bounded Barron norm and conclude that functions in the Barron space could
be learnt from their random samples efficiently.

该研究讨论了基于图的卷积神经网络（GCNN）在 Barron 函数空间中的应用，证明了该空间是一个再生核 Banach 空间，可以通过 GCNN 的输出来有效逼近。同时估计了有界 Barron 范数函数的 Rademacher 复杂度，并指出可以有效地从随机样本中学习 Barron 空间中的函数。

Barron 空间用于图卷积神经网络

Barron Space for Graph Convolution Neural Networks

Sparsity of a learning solution is a desirable feature in machine learning.
Certain reproducing kernel Banach spaces (RKBSs) are appropriate hypothesis
spaces for sparse learning methods. The goal of this paper is to understand
what kind of RKBSs can promote sparsity for learning solutions. We consider two
typical learning models in an RKBS: the minimum norm interpolation (MNI)
problem and the regularization problem. We first establish an explicit
representer theorem for solutions of these problems, which represents the
extreme points of the solution set by a linear combination of the extreme
points of the subdifferential set, of the norm function, which is
data-dependent. We then propose sufficient conditions on the RKBS that can
transform the explicit representation of the solutions to a sparse kernel
representation having fewer terms than the number of the observed data. Under
the proposed sufficient conditions, we investigate the role of the
regularization parameter on sparsity of the regularized solutions. We further
show that two specific RKBSs: the sequence space $\ell_1(\mathbb{N})$ and the
measure space can have sparse representer theorems for both MNI and
regularization models.

通过在 Reproducing Kernel Banach Space（RKBS）中建立显式的代表定理，我们探讨了该 RKBS 中代表的性质，与正则化参数对于解决方案的稀疏性的关系，并证明了一些特定的 RKBS 具有解决稀疏学习问题的能力