Convolutional neural networks are the most widely used type of neural networks in applications. In mathematical analysis, however, mostly fully-connected networks are studied. In this paper, we establish a connection between both network architectures. Using this connection, we show that all upper and lower bounds concerning approximation rates of {fully-connected} neural networks for functions $f \in \mathcal{C}$ -- for an arbitrary function class $\mathcal{C}$ -- translate to essentially the same bounds concerning approximation rates of convolutional neural networks for functions $f \in {\mathcal{C}^{equi}}$, with the class ${\mathcal{C}^{equi}}$ consisting of all translation equivariant functions whose first coordinate belongs to $\mathcal{C}$. All presented results consider exclusively the case of convolutional neural networks without any pooling operation and with circular convolutions, i.e., not based on zero-padding.
翻译:进化神经网络是应用中最广泛使用的神经网络类型。 然而, 在数学分析中, 研究的大多是完全连接的网络。 在本文中, 我们建立两个网络结构之间的连接。 使用此连接, 我们显示, 功能 $f\ in\ mathcal{C} $ 的 神经网络近似速率的所有上限和下限 。 对于任意函数级 $\ mathcal{C} $ -- 翻译为关于函数 $f \ in ~ mathcal{C ⁇ equi $ 的进化神经网络近似速率的界限, 与 $\ mathcal{C equi $ 类的 包含所有翻译等同功能的上限 $\ mathcal{ C} $ 。 所有显示的结果都只考虑进化神经网络的情况, 没有集中操作, 循环变相, 即不以零平板为基础 。