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

Journal Article

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 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 analytical results are validated by numerical experiments on the approximation of a 1D Fourier kernel and of the energy of 1D and 2D Poisson's equations. Butterfly-net with trained parameters outperforms the hard-coded Butterfly-net and achieves similar accuracy as the trained CNN but with much less parameters. In addition, better robustness of Butterfly-net against CNN is demonstrated when the distribution of the input data has domain shift.

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Duke Authors

Cited Authors

  • Cheng, X; Li, Y; Lu, J