# Dimension Mixer: Group Mixing of Input Dimensions for Efficient Function Approximation

arXiv: https://arxiv.org/abs/2311.18735 · HTML: https://mlsanity.com/paper/2311.18735/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 61 of its 72 references that are in the map.
- First posted on arXiv 30 Nov 2023.

## Builds on (most cited first, 30 of 61)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [Very Deep Convolutional Networks for Large-Scale Image Recognition](https://mlsanity.com/paper/1409.1556/) (2014)
- [Swin Transformer: Hierarchical Vision Transformer using Shifted Windows](https://mlsanity.com/paper/2103.14030/) (2021)
- [MobileNets: Efficient Convolutional Neural Networks for Mobile Vision Applications](https://mlsanity.com/paper/1704.04861/) (2017)
- [FlashAttention: Fast and Memory-Efficient Exact Attention with IO-Awareness](https://mlsanity.com/paper/2205.14135/) (2022)
- [Longformer: The Long-Document Transformer](https://mlsanity.com/paper/2004.05150/) (2020)
- [Switch Transformers: Scaling to Trillion Parameter Models with Simple and Efficient Sparsity](https://mlsanity.com/paper/2101.03961/) (2021)
- [Xception: Deep Learning with Depthwise Separable Convolutions](https://mlsanity.com/paper/1610.02357/) (2016)
- [Learning both Weights and Connections for Efficient Neural Networks](https://mlsanity.com/paper/1506.02626/) (2015)
- [Megatron-LM: Training Multi-Billion Parameter Language Models Using Model Parallelism](https://mlsanity.com/paper/1909.08053/) (2019)
- [Reformer: The Efficient Transformer](https://mlsanity.com/paper/2001.04451/) (2020)
- [Rethinking Attention with Performers](https://mlsanity.com/paper/2009.14794/) (2020)
- [Generating Long Sequences with Sparse Transformers](https://mlsanity.com/paper/1904.10509/) (2019)
- [Glow: Generative Flow with Invertible 1x1 Convolutions](https://mlsanity.com/paper/1807.03039/) (2018)
- [Linformer: Self-Attention with Linear Complexity](https://mlsanity.com/paper/2006.04768/) (2020)
- [MLP-Mixer: An all-MLP Architecture for Vision](https://mlsanity.com/paper/2105.01601/) (2021)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [ShuffleNet V2: Practical Guidelines for Efficient CNN Architecture Design](https://mlsanity.com/paper/1807.11164/) (2018)
- [EfficientNetV2: Smaller Models and Faster Training](https://mlsanity.com/paper/2104.00298/) (2021)
- [NICE: Non-linear Independent Components Estimation](https://mlsanity.com/paper/1410.8516/) (2014)
- [Geometric deep learning: going beyond Euclidean data](https://mlsanity.com/paper/1611.08097/) (2016)
- [Efficient Transformers: A Survey](https://mlsanity.com/paper/2009.06732/) (2020)
- [Long Range Arena: A Benchmark for Efficient Transformers](https://mlsanity.com/paper/2011.04006/) (2020)
- [MaxViT: Multi-Axis Vision Transformer](https://mlsanity.com/paper/2204.01697/) (2022)
- [Nystr"omformer: A Nystr"om-Based Algorithm for Approximating Self-Attention](https://mlsanity.com/paper/2102.03902/) (2021)
- [Pay Attention to MLPs](https://mlsanity.com/paper/2105.08050/) (2021)
- [Flow++: Improving Flow-Based Generative Models with Variational Dequantization and Architecture Design](https://mlsanity.com/paper/1902.00275/) (2019)
- [Neighborhood Attention Transformer](https://mlsanity.com/paper/2204.07143/) (2022)

## Abstract (arXiv)

The recent success of multiple neural architectures like CNNs, Transformers, and MLP-Mixers motivated us to look for similarities and differences between them. We found that these architectures can be interpreted through the lens of a general concept of dimension mixing. Research on coupling flows and the butterfly transform shows that partial and hierarchical signal mixing schemes are sufficient for efficient and expressive function approximation. In this work, we study group-wise sparse, non-linear, multi-layered and learnable mixing schemes of inputs and find that they are complementary to many standard neural architectures. Following our observations and drawing inspiration from the Fast Fourier Transform, we generalize Butterfly Structure to use non-linear mixer function allowing for MLP as mixing function called Butterfly MLP. We were also able to sparsely mix along sequence dimension for Transformer-based architectures called Butterfly Attention. Experiments on CIFAR and LRA datasets demonstrate that the proposed Non-Linear Butterfly Mixers are efficient and scale well when the host architectures are used as mixing function. Additionally, we propose Patch-Only MLP-Mixer for processing spatial 2D signals demonstrating a different dimension mixing strategy.