  Deep neural networks (DNNs) generate much richer function spaces than shallow
networks. Since the function spaces induced by shallow networks have several
approximation theoretic drawbacks, this explains, however, not necessarily the
success of deep networks. In this article we take another route by comparing
the expressive power of DNNs with ReLU activation function to piecewise linear
spline methods. We show that MARS (multivariate adaptive regression splines) is
improper learnable by DNNs in the sense that for any given function that can be
expressed as a function in MARS with $M$ parameters there exists a multilayer
neural network with $O(M \log (M/\varepsilon))$ parameters that approximates
this function up to sup-norm error $\varepsilon.$ We show a similar result for
expansions with respect to the Faber-Schauder system. Based on this, we derive
risk comparison inequalities that bound the statistical risk of fitting a
neural network by the statistical risk of spline-based methods. This shows that
deep networks perform better or only slightly worse than the considered spline
methods. We provide a constructive proof for the function approximations.
