# Beyond the Permutation Symmetry of Transformers: The Role of Rotation for Model Fusion

arXiv: https://arxiv.org/abs/2502.00264 · HTML: https://mlsanity.com/paper/2502.00264/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- Builds on 72 of its 100 references that are in the map.
- First posted on arXiv 1 Feb 2025.

## Builds on (most cited first, 30 of 72)

- [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)
- [Learning Transferable Visual Models From Natural Language Supervision](https://mlsanity.com/paper/2103.00020/) (2021)
- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [Exploring the Limits of Transfer Learning with a Unified Text-to-Text Transformer](https://mlsanity.com/paper/1910.10683/) (2019)
- [RoBERTa: A Robustly Optimized BERT Pretraining Approach](https://mlsanity.com/paper/1907.11692/) (2019)
- [Swin Transformer: Hierarchical Vision Transformer using Shifted Windows](https://mlsanity.com/paper/2103.14030/) (2021)
- [Communication-Efficient Learning of Deep Networks from Decentralized Data](https://mlsanity.com/paper/1602.05629/) (2016)
- [BART: Denoising Sequence-to-Sequence Pre-training for Natural Language Generation, Translation, and Comprehension](https://mlsanity.com/paper/1910.13461/) (2019)
- [Layer Normalization](https://mlsanity.com/paper/1607.06450/) (2016)
- [GLUE: A Multi-Task Benchmark and Analysis Platform for Natural Language Understanding](https://mlsanity.com/paper/1804.07461/) (2018)
- [Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles](https://mlsanity.com/paper/1612.01474/) (2016)
- [DeBERTa: Decoding-enhanced BERT with Disentangled Attention](https://mlsanity.com/paper/2006.03654/) (2020)
- [Federated Learning: Strategies for Improving Communication Efficiency](https://mlsanity.com/paper/1610.05492/) (2016)
- [Model soups: averaging weights of multiple fine-tuned models improves accuracy without increasing inference time](https://mlsanity.com/paper/2203.05482/) (2022)
- [Averaging Weights Leads to Wider Optima and Better Generalization](https://mlsanity.com/paper/1803.05407/) (2018)
- [TIES-Merging: Resolving Interference When Merging Models](https://mlsanity.com/paper/2306.01708/) (2023)
- [Merging Models with Fisher-Weighted Averaging](https://mlsanity.com/paper/2111.09832/) (2021)
- [Federated Learning with Matched Averaging](https://mlsanity.com/paper/2002.06440/) (2020)
- [Graph Transformer Networks](https://mlsanity.com/paper/1911.06455/) (2019)
- [Git Re-Basin: Merging Models modulo Permutation Symmetries](https://mlsanity.com/paper/2209.04836/) (2022)
- [SWAD: Domain Generalization by Seeking Flat Minima](https://mlsanity.com/paper/2102.08604/) (2021)
- [Bayesian Nonparametric Federated Learning of Neural Networks](https://mlsanity.com/paper/1905.12022/) (2019)
- [Dataless Knowledge Fusion by Merging Weights of Language Models](https://mlsanity.com/paper/2212.09849/) (2022)
- [The Role of Permutation Invariance in Linear Mode Connectivity of Neural Networks](https://mlsanity.com/paper/2110.06296/) (2021)
- [AdaMerging: Adaptive Model Merging for Multi-Task Learning](https://mlsanity.com/paper/2310.02575/) (2023)
- [Model Fusion via Optimal Transport](https://mlsanity.com/paper/1910.05653/) (2019)
- [Branch-Train-Merge: Embarrassingly Parallel Training of Expert Language Models](https://mlsanity.com/paper/2208.03306/) (2022)
- [Algorithmic Regularization in Learning Deep Homogeneous Models: Layers are Automatically Balanced](https://mlsanity.com/paper/1806.00900/) (2018)

## Built on by (most cited first, 16 of 16)

- [Symmetry in Neural Network Parameter Spaces](https://mlsanity.com/paper/2506.13018/) (2025)
- [Model Unmerging: Making Your Models Unmergeable for Secure Model Sharing](https://mlsanity.com/paper/2509.01548/) (2025)
- [Symmetry Breaking in Transformers for Efficient and Interpretable Training](https://mlsanity.com/paper/2601.22257/) (2026)
- [Label-Free Cross-Task LoRA Merging with Null-Space Compression](https://mlsanity.com/paper/2603.26317/) (2026)
- [Toward Manifest Relationality in Transformers via Symmetry Reduction](https://mlsanity.com/paper/2602.18948/) (2026)
- [Preference-Aligned LoRA Merging: Preserving Subspace Coverage and Addressing Directional Anisotropy](https://mlsanity.com/paper/2603.26299/) (2026)
- [MOMO: Mars Orbital Model Foundation Model for Mars Orbital Applications](https://mlsanity.com/paper/2604.02719/) (2026)
- [Scaling Linear Mode Connectivity and Merging to Billion Parameter Pretrained Transformers](https://mlsanity.com/paper/2606.23607/) (2026)
- [Generalizing the Geometry of Model Merging Through Frechet Averages](https://mlsanity.com/paper/2604.27155/) (2026)
- [Symmetry-Aware Graph Metanetwork Autoencoders: Model Merging through Parameter Canonicalization](https://mlsanity.com/paper/2511.12601/) (2025)
- [The Role of Symmetry in Optimizing Overparameterized Networks](https://mlsanity.com/paper/2604.25150/) (2026)
- [Deep neural networks as lattice gauge theories](https://mlsanity.com/paper/2608.19331/) (2026)
- [When Privacy Hurts Mergeability: Geometry-Aware Model Merging under Differential Privacy](https://mlsanity.com/paper/2608.26655/) (2026)
- [Multi-Head Self Attention is a Parameter Identification Mechanism](https://mlsanity.com/paper/2609.01231/) (2026)
- [The Linear Representation Hypothesis Needs a Group Action](https://mlsanity.com/paper/2609.27158/) (2026)
- [Discovering Symmetries in Neural Network Parameter Spaces](https://mlsanity.com/paper/2609.33527/) (2026)

## Abstract (arXiv)

Symmetry in the parameter space of deep neural networks (DNNs) has proven beneficial for various deep learning applications. A well-known example is the permutation symmetry in Multi-Layer Perceptrons (MLPs), where permuting the rows of weight matrices in one layer and applying the inverse permutation to adjacent layers yields a functionally equivalent model. While permutation symmetry fully characterizes the equivalence set for MLPs, its discrete nature limits its utility for transformers. In this paper, we introduce rotation symmetry, a novel form of parameter space symmetry for transformers that generalizes permutation symmetry by rotating parameter matrices in self-attention layers. Unlike permutation symmetry, rotation symmetry operates in a continuous domain, thereby significantly expanding the equivalence set for transformers. Based on this property, we propose a theoretically optimal parameter matching algorithm as a plug-and-play module to enhance model fusion. We evaluate our approach using pre-trained transformers across diverse natural language and vision tasks. Experimental results demonstrate that our rotation symmetry-based matching algorithm substantially improves model fusion, highlighting the potential of parameter space symmetry to facilitate model fusion. Our code is available on https://github.com/zhengzaiyi/RotationSymmetry.