# Flopping for FLOPs: Leveraging equivariance for computational efficiency

arXiv: https://arxiv.org/abs/2502.05169 · HTML: https://mlsanity.com/paper/2502.05169/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 47 of its 66 references that are in the map.
- First posted on arXiv 7 Feb 2025.

## Builds on (most cited first, 30 of 47)

- [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)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [ImageNet Large Scale Visual Recognition Challenge](https://mlsanity.com/paper/1409.0575/) (2014)
- [Scaling Laws for Neural Language Models](https://mlsanity.com/paper/2001.08361/) (2020)
- [Layer Normalization](https://mlsanity.com/paper/1607.06450/) (2016)
- [Neural Machine Translation by Jointly Learning to Align and Translate](https://mlsanity.com/paper/1409.0473/) (2014)
- [A ConvNet for the 2020s](https://mlsanity.com/paper/2201.03545/) (2022)
- [Gaussian Error Linear Units (GELUs)](https://mlsanity.com/paper/1606.08415/) (2016)
- [FlashAttention-2: Faster Attention with Better Parallelism and Work Partitioning](https://mlsanity.com/paper/2307.08691/) (2023)
- [Identity Mappings in Deep Residual Networks](https://mlsanity.com/paper/1603.05027/) (2016)
- [Do ImageNet Classifiers Generalize to ImageNet?](https://mlsanity.com/paper/1902.10811/) (2019)
- [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)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Going deeper with Image Transformers](https://mlsanity.com/paper/2103.17239/) (2021)
- [General E(2)-Equivariant Steerable CNNs](https://mlsanity.com/paper/1911.08251/) (2019)
- [DeiT III: Revenge of the ViT](https://mlsanity.com/paper/2204.07118/) (2022)
- [ResMLP: Feedforward networks for image classification with data-efficient training](https://mlsanity.com/paper/2105.03404/) (2021)
- [Steerable CNNs](https://mlsanity.com/paper/1612.08498/) (2016)
- [Understanding image representations by measuring their equivariance and equivalence](https://mlsanity.com/paper/1411.5908/) (2014)
- [Training Very Deep Networks](https://mlsanity.com/paper/1507.06228/) (2015)
- [Clebsch-Gordan Nets: a Fully Fourier Space Spherical Convolutional Neural Network](https://mlsanity.com/paper/1806.09231/) (2018)
- [Roto-Translation Covariant Convolutional Networks for Medical Image Analysis](https://mlsanity.com/paper/1804.03393/) (2018)
- [Exploiting Cyclic Symmetry in Convolutional Neural Networks](https://mlsanity.com/paper/1602.02660/) (2016)
- [Learning and Leveraging World Models in Visual Representation Learning](https://mlsanity.com/paper/2403.00504/) (2024)
- [Geometric Deep Learning and Equivariant Neural Networks](https://mlsanity.com/paper/2105.13926/) (2021)
- [Attentive Group Equivariant Convolutional Networks](https://mlsanity.com/paper/2002.03830/) (2020)
- [Do You Even Need Attention? A Stack of Feed-Forward Layers Does Surprisingly Well on ImageNet](https://mlsanity.com/paper/2105.02723/) (2021)

## Built on by (most cited first, 8 of 8)

- [LoMa: Local Feature Matching Revisited](https://mlsanity.com/paper/2604.04931/) (2026)
- [Platonic Transformers: A Solid Choice For Equivariance](https://mlsanity.com/paper/2510.03511/) (2025)
- [Quick ViTs: Speeding up Vision Transformers through Equivariance](https://mlsanity.com/paper/2505.15441/) (2025)
- [Vanilla Group Equivariant Vision Transformer: Simple and Effective](https://mlsanity.com/paper/2602.08047/) (2026)
- [Who Handles Orientation? Investigating Invariance in Feature Matching](https://mlsanity.com/paper/2604.11809/) (2026)
- [Tunable Soft Equivariance with Guarantees](https://mlsanity.com/paper/2603.26657/) (2026)
- [A Unified Framework for Vision Transformers Equivariant to Discrete Subgroups of O(2)](https://mlsanity.com/paper/2606.27864/) (2026)
- [Flash EQ-Linear: Accelerating Equivariant Linear Layers via Group-wise Discrete Fourier Transform](https://mlsanity.com/paper/2607.21271/) (2026)

## Abstract (arXiv)

Incorporating geometric invariance into neural networks enhances parameter efficiency but typically increases computational costs. This paper introduces new equivariant neural networks that preserve symmetry while maintaining a comparable number of floating-point operations (FLOPs) per parameter to standard non-equivariant networks. We focus on horizontal mirroring (flopping) invariance, common in many computer vision tasks. The main idea is to parametrize the feature spaces in terms of mirror-symmetric and mirror-antisymmetric features, i.e., irreps of the flopping group. This decomposes the linear layers to be block-diagonal, requiring half the number of FLOPs. Our approach reduces both FLOPs and wall-clock time, providing a practical solution for efficient, scalable symmetry-aware architectures.