Butterfly-Net: Optimal Function Representation Based on Convolutional Neural Networks

Deep networks, especially Convolutional Neural Networks (CNNs), have been successfully applied in various areas of machine learning as well as to challenging problems in other scientific and engineering fields. This paper introduces Butterfly-Net, a low-complexity CNN with structured hard-coded weights and sparse across-channel connections, which aims at an optimal hierarchical function representation of the input signal. Theoretical analysis of the approximation power of Butterfly-Net to the Fourier representation of input data shows that the error decays exponentially as the depth increases. Due to the ability of Butterfly-Net to approximate Fourier and local Fourier transforms, the result can be used for approximation upper bound for CNNs in a large class of problems. The analysis results are validated in numerical experiments on the approximation of a 1D Fourier kernel and of solving a 2D Poisson's equation.