# Sparse Linear Networks with a Fixed Butterfly Structure: Theory and Practice

arXiv: https://arxiv.org/abs/2007.08864 · HTML: https://mlsanity.com/paper/2007.08864/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Jul 2020.

## Built on by (most cited first, 9 of 9)

- [Monarch: Expressive Structured Matrices for Efficient and Accurate Training](https://mlsanity.com/paper/2204.00595/) (2022)
- [Simple Hardware-Efficient Long Convolutions for Sequence Modeling](https://mlsanity.com/paper/2302.06646/) (2023)
- [Generalization Bounds for Data-Driven Numerical Linear Algebra](https://mlsanity.com/paper/2206.07886/) (2022)
- [Rethinking Neural Operations for Diverse Tasks](https://mlsanity.com/paper/2103.15798/) (2021)
- [Sparse Training Theory for Scalable and Efficient Agents](https://mlsanity.com/paper/2103.01636/) (2021)
- [Algorithm Configuration for Structured Pfaffian Settings](https://mlsanity.com/paper/2409.04367/) (2024)
- [Improved Generalization Bound and Learning of Sparsity Patterns for Data-Driven Low-Rank Approximation](https://mlsanity.com/paper/2209.08281/) (2022)
- [Efficient NTK using Dimensionality Reduction](https://mlsanity.com/paper/2210.04807/) (2022)
- [Learning Sparsity and Randomness for Data-driven Low Rank Approximation](https://mlsanity.com/paper/2212.08186/) (2022)

## Abstract (arXiv)

A butterfly network consists of logarithmically many layers, each with a linear number of non-zero weights (pre-specified). The fast Johnson-Lindenstrauss transform (FJLT) can be represented as a butterfly network followed by a projection onto a random subset of the coordinates. Moreover, a random matrix based on FJLT with high probability approximates the action of any matrix on a vector. Motivated by these facts, we propose to replace a dense linear layer in any neural network by an architecture based on the butterfly network. The proposed architecture significantly improves upon the quadratic number of weights required in a standard dense layer to nearly linear with little compromise in expressibility of the resulting operator. In a collection of wide variety of experiments, including supervised prediction on both the NLP and vision data, we show that this not only produces results that match and at times outperform existing well-known architectures, but it also offers faster training and prediction in deployment. To understand the optimization problems posed by neural networks with a butterfly network, we also study the optimization landscape of the encoder-decoder network, where the encoder is replaced by a butterfly network followed by a dense linear layer in smaller dimension. Theoretical result presented in the paper explains why the training speed and outcome are not compromised by our proposed approach.