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

Besov Function Approximation and Binary Classification on Low-Dimensional Manifolds Using Convolutional Residual Networks