Optimal Approximation with Sparsely Connected Deep Neural Networks

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 systemsso-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. Our central 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 $\alpha^{{-1}$-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. Moreover, these results indicate that stochastic gradient descent can actually learn approximations that are sparse in the representation systems optimally sparsifying the function class the network is trained on.