  Deep learning based on deep neural networks has been very successful in many
practical applications, but it lacks enough theoretical understanding due to
the network architectures and structures. In this paper we establish some
analysis for linear feature extraction by a deep multi-channel convolutional
neural networks (CNNs), which demonstrates the power of deep learning over
traditional linear transformations, like Fourier, wavelets, redundant
dictionary coding methods. Moreover, we give an exact construction presenting
how linear features extraction can be conducted efficiently with multi-channel
CNNs. It can be applied to lower the essential dimension for approximating a
high dimensional function. Rates of function approximation by such deep
networks implemented with channels and followed by fully-connected layers are
investigated as well. Harmonic analysis for factorizing linear features into
multi-resolution convolutions plays an essential role in our work.
Nevertheless, a dedicate vectorization of matrices is constructed, which
bridges 1D CNN and 2D CNN and allows us to have corresponding 2D analysis.
