Approximation analysis of CNNs from a feature extraction view

TL;DR

This paper provides a theoretical analysis of how deep multi-channel CNNs can efficiently perform linear feature extraction, demonstrating their superiority over traditional methods and exploring their approximation capabilities.

Contribution

It establishes a theoretical framework for linear feature extraction using CNNs, including an exact construction and analysis of approximation rates, bridging 1D and 2D CNNs.

Findings

Deep CNNs outperform traditional linear methods in feature extraction.

An exact construction for efficient linear feature extraction with CNNs is provided.

Analysis of approximation rates for high-dimensional functions using CNNs.

Abstract

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.