论文标题
深卷积神经网络的理论II：球形分析
Theory of Deep Convolutional Neural Networks II: Spherical Analysis
论文作者
论文摘要
基于各种结构和架构的深度神经网络的深度学习在许多实际应用中都很强大，但缺乏足够的理论验证。在本文中，我们考虑了一个深度卷积神经网络的家族，应用于单位球的近似功能$ \ mathbb {s}^{d-1} $ $ \ mathbb {r}^d $。当近似函数位于sobolev空间中$ w^r_ \ infty（\ mathbb {s}^{d-1}）$带有$ r> 0 $时，我们的分析呈现均匀近似的速率。我们的工作从理论上验证了深卷积神经网络的建模和近似能力，然后进行下采样和一两个完全连接的层。球形分析的关键思想是使用球形谐波空间的再现核的内部产品形式，然后应用过滤器的卷积因子化来实现生成的线性特征。
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 $W^r_\infty (\mathbb{S}^{d-1})$ 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.
