Approximation and non-parametric estimation of functions over high-dimensional spheres via deep ReLU networks

We develop a new approximation and estimation analysis of deep feed-forward neural networks (FNNs) with the Rectified Linear Unit (ReLU) activation. The functions of interests for the approximation and estimation are assumed to be from Sobolev spaces defined over the $d$-dimensional unit sphere with smoothness index $r>0$. In the regime where $r$ is in the constant order (i.e., $r=\mathcal{O}(1)$), it is shown that at most $d^d$ active parameters are required for getting $d^{-C}$ approximation rate for some constant $C>0$. In contrast, in the regime where the index $r$ grows in the order of $d$ (i.e., $r=\mathcal{O}(d)$) asymptotically, we prove the approximation error decays in the rate $d^{-d^{\beta}}$ with $0<\beta<1$ up to some constant factor independent of $d$. The required number of active parameters in the networks for the approximation increases polynomially in $d$ as $d\rightarrow{\infty}$. In addition to this, it is shown that bound on the excess risk has a $d^d$ factor, when $r=\mathcal{O}(1)$, whereas it has $d^{\mathcal{O}(1)}$ factor, when $r=\mathcal{O}(d)$. We emphasize our findings by making comparisons to the results on approximation and estimation errors of deep ReLU FNN when functions are from Sobolev spaces defined over $d$-dimensional cube. Here, we show that with the current state-of-the-art result, $d^{d}$ factor remain both in the approximation and estimation error, regardless of the order of $r$.