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

@article{Li2020ButterflyNetOF, title={Butterfly-Net: Optimal Function Representation Based on Convolutional Neural Networks}, author={Yingzhou Li and Xiuyuan Cheng and Jianfeng Lu}, journal={ArXiv}, year={2020}, volume={abs/1805.07451} }

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…

## 17 Citations

Butterfly Transform: An Efficient FFT Based Neural Architecture Design

- Computer Science2020 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR)
- 2020

It is shown that extending the butterfly operations from the FFT algorithm to a general Butterfly Transform (BFT) can be beneficial in building an efficient block structure for CNN designs, and ShuffleNet-V2+BFT outperforms state-of-the-art architecture search methods MNasNet, FBNet and MobilenetV3 in the low FLOP regime.

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- Computer ScienceArXiv
- 2020

An end-to-end deep learning architecture called the wide-band butterfly network (WideBNet) is introduced, which incorporates tools from computational harmonic analysis and traditional multi-scale methods to drastically reduce the number of trainable parameters to match the inherent complexity of the problem.

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- Computer ScienceArXiv
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A novel deep learning framework based on integral autoencoders (IAE-Net) for discretization invariant learning that achieves state-of-the-art performance in existing applications and creates a wide range of new applications where existing methods fail.

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- Computer ScienceResearch in the Mathematical Sciences
- 2019

A multiscale artificial neural network for high-dimensional nonlinear maps based on the idea of hierarchical nested bases in the fast multipole method and the H2-matrices to efficiently approximate discretized non linear maps arising from partial differential equations or integral equations.

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- Computer ScienceICLR
- 2020

A family of matrices called kaleidoscope matrices (K-matrices) are introduced that provably capture any structured matrix with near-optimal space (parameter) and time (arithmetic operation) complexity that can be automatically learned within end-to-end pipelines to replace hand-crafted procedures.

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A novel neural network architecture, SwitchNet, is proposed for solving the wave equation based inverse scattering problems via providing maps between the scatterers and the scattered field (and vice versa) by leveraging the inherent low-rank structure of the scattering problems and introducing a novel switching layer with sparse connections.

Accurate and Robust Deep Learning Framework for Solving Wave-Based Inverse Problems in the Super-Resolution Regime

- Computer ScienceArXiv
- 2021

An end-to-end deep learning framework that comprehensively solves the inverse wave scattering problem across all length scales and outperforms both classical inversion and competing network architectures that specialize in oscillatory wave scattering data is proposed.

BCR-Net: a neural network based on the nonstandard wavelet form

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Variational training of neural network approximations of solution maps for physical models

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A reduced order Schwarz method for nonlinear multiscale elliptic equations based on two-layer neural networks

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A solver for multiscale fully nonlinear elliptic equations that makes use of domain decomposition, an accelerated Schwarz framework, and two-layer neural networks to approximate the boundary-to-boundary map for the subdomains, which is the key step in the Schwarz procedure.

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