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.

翻译：基于深度神经网络的深度学习已在许多实际应用中取得巨大成功，但由于网络架构和结构的复杂性，其理论理解仍存在不足。本文针对深度多通道卷积神经网络在线性特征提取中的表现建立了分析框架，揭示了深度学习相较于傅里叶变换、小波变换及冗余字典编码等传统线性变换的优越性。我们进一步提出了一种精确的构造方法，展示了如何利用多通道卷积神经网络高效实现线性特征提取，该方法可有效降低高维函数逼近所需的本征维度。同时研究了此类由卷积层与全连接层级联构成的深度网络在函数逼近中的收敛速率。将线性特征分解为多分辨率卷积的调和分析是本文工作的核心基础。此外，通过构造精巧的矩阵向量化方法，我们建立了1D CNN与2D CNN之间的关联，从而实现了相应的二维分析。