  Most of existing statistical theories on deep neural networks have sample
complexities cursed by the data dimension and therefore cannot well explain the
empirical success of deep learning on high-dimensional data. To bridge this
gap, we propose to exploit low-dimensional geometric structures of the real
world data sets. We establish theoretical guarantees of convolutional residual
networks (ConvResNet) in terms of function approximation and statistical
estimation for binary classification. Specifically, given the data lying on a
$d$-dimensional manifold isometrically embedded in $\mathbb{R}^D$, we prove
that if the network architecture is properly chosen, ConvResNets can (1)
approximate Besov functions on manifolds with arbitrary accuracy, and (2) learn
a classifier by minimizing the empirical logistic risk, which gives an excess
risk in the order of $n^{-\frac{s}{2s+2(s\vee d)}}$, where $s$ is a smoothness
parameter. This implies that the sample complexity depends on the intrinsic
dimension $d$, instead of the data dimension $D$. Our results demonstrate that
ConvResNets are adaptive to low-dimensional structures of data sets.
