We derive fundamental lower bounds on the connectivity and the memory requirements of deep neural networks guaranteeing uniform approximation rates for arbitrary function classes in $L^2(\mathbb{R}^d)$. In other words, we establish a connection between the complexity of a function class and the complexity of deep neural networks approximating functions from this class to within a prescribed accuracy. Additionally, we prove that our lower bounds are achievable for a broad family of function classes. Specifically, all function classes that are optimally approximated by a general class of representation systems---so-called \emph{affine systems}---can be approximated by deep neural networks with minimal connectivity and memory requirements. Affine systems encompass a wealth of representation systems from applied harmonic analysis such as wavelets, ridgelets, curvelets, shearlets, $\alpha$-shearlets, and more generally $\alpha$-molecules. This result elucidates a remarkable universality property of neural networks and shows that they achieve the optimum approximation properties of all affine systems combined. As a specific example, we consider the class of $1/\alpha$-cartoon-like functions, which is approximated optimally by $\alpha$-shearlets. We also explain how our results can be extended to the case of functions on low-dimensional immersed manifolds. Finally, we present numerical experiments demonstrating that the standard stochastic gradient descent algorithm generates deep neural networks providing close-to-optimal approximation rates at minimal connectivity. Moreover, these results show that stochastic gradient descent actually learns approximations that are sparse in the representation systems optimally sparsifying the function class the network is trained on.

本文提出了深度神经网络的可连接性和内存需求的基本下限，同时证明了其实现方式适用于广泛的函数类。此外，研究表明，广义仿射系统内的全局极优逼近问题可以通过神经网络得到最优解，并通过数值实验验证了随机梯度下降算法能够学习出近乎最优的函数逼近。

稀疏连接深度神经网络的最优逼近