Deep learning based on deep neural networks of various structures and architectures has been powerful in many practical applications, but it lacks enough theoretical verifications. In this paper, we consider a family of deep convolutional neural networks applied to approximate functions on the unit sphere ${\mathbb{S}}^{d-1}$ of ${\mathbb{R}}^d$. Our analysis presents rates of uniform approximation when the approximated function lies in the Sobolev space $W_{\infty }^r\left({\mathbb{S}}^{d-1}\right)$ with $r>0$ or takes an additive ridge form. Our work verifies theoretically the modelling and approximation ability of deep convolutional neural networks followed by downsampling and one fully connected layer or two. The key idea of our spherical analysis is to use the inner product form of the reproducing kernels of the spaces of spherical harmonics and then to apply convolutional factorizations of filters to realize the generated linear features.

基于各种结构和体系结构的深度神经网络的深度学习在许多实际应用中功能强大，但缺乏足够的理论验证。在本文中，我们考虑了一系列深卷积神经网络，它们适用于单位球面上的近似函数${\mathbb{小号}}^{d-\mathrm{1个}}$ 的 ${\mathbb{[R}}^d$。当逼近函数位于Sobolev空间中时，我们的分析给出了统一逼近率${\mathrm{w\; ^}}_{\infty }^{\mathrm{[R}}（{\mathbb{小号}}^{d-\mathrm{1个}}）$ 与 $\mathrm{[R}>0$或采用附加凸脊形式。我们的工作从理论上验证了深度卷积神经网络的建模和逼近能力，然后进行了下采样和一层或两层完全连接。我们球面分析的关键思想是使用球面谐波空间的再生核的内积形式，然后应用滤波器的卷积因式分解来实现生成的线性特征。