# The Role of Permutation Invariance in Linear Mode Connectivity of Neural Networks

arXiv: https://arxiv.org/abs/2110.06296 · HTML: https://mlsanity.com/paper/2110.06296/

## Key facts

- Cited by 291 papers in the map (data through 14 Oct 2026).
- Builds on 34 of its 45 references that are in the map.
- First posted on arXiv 12 Oct 2021.

## Builds on (most cited first, 30 of 34)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Very Deep Convolutional Networks for Large-Scale Image Recognition](https://mlsanity.com/paper/1409.1556/) (2014)
- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [The Lottery Ticket Hypothesis: Finding Sparse, Trainable Neural Networks](https://mlsanity.com/paper/1803.03635/) (2018)
- [Understanding deep learning requires rethinking generalization](https://mlsanity.com/paper/1611.03530/) (2016)
- [On Large-Batch Training for Deep Learning: Generalization Gap and Sharp Minima](https://mlsanity.com/paper/1609.04836/) (2016)
- [Averaging Weights Leads to Wider Optima and Better Generalization](https://mlsanity.com/paper/1803.05407/) (2018)
- [Visualizing the Loss Landscape of Neural Nets](https://mlsanity.com/paper/1712.09913/) (2017)
- [Reconciling modern machine learning practice and the bias-variance trade-off](https://mlsanity.com/paper/1812.11118/) (2018)
- [Deep Double Descent: Where Bigger Models and More Data Hurt](https://mlsanity.com/paper/1912.02292/) (2019)
- [Linear Mode Connectivity and the Lottery Ticket Hypothesis](https://mlsanity.com/paper/1912.05671/) (2019)
- [Exploring Generalization in Deep Learning](https://mlsanity.com/paper/1706.08947/) (2017)
- [Loss Surfaces, Mode Connectivity, and Fast Ensembling of DNNs](https://mlsanity.com/paper/1802.10026/) (2018)
- [A Mean Field View of the Landscape of Two-Layers Neural Networks](https://mlsanity.com/paper/1804.06561/) (2018)
- [Deep Ensembles: A Loss Landscape Perspective](https://mlsanity.com/paper/1912.02757/) (2019)
- [BatchEnsemble: An Alternative Approach to Efficient Ensemble and Lifelong Learning](https://mlsanity.com/paper/2002.06715/) (2020)
- [What is being transferred in transfer learning?](https://mlsanity.com/paper/2008.11687/) (2020)
- [Essentially No Barriers in Neural Network Energy Landscape](https://mlsanity.com/paper/1803.00885/) (2018)
- [Loss landscapes and optimization in over-parameterized non-linear systems and neural networks](https://mlsanity.com/paper/2003.00307/) (2020)
- [Uniform convergence may be unable to explain generalization in deep learning](https://mlsanity.com/paper/1902.04742/) (2019)
- [Model Fusion via Optimal Transport](https://mlsanity.com/paper/1910.05653/) (2019)
- [Geometry of the Loss Landscape in Overparameterized Neural Networks: Symmetries and Invariances](https://mlsanity.com/paper/2105.12221/) (2021)
- [Path-SGD: Path-Normalized Optimization in Deep Neural Networks](https://mlsanity.com/paper/1506.02617/) (2015)
- [Topology and Geometry of Half-Rectified Network Optimization](https://mlsanity.com/paper/1611.01540/) (2016)
- [Optimizing Mode Connectivity via Neuron Alignment](https://mlsanity.com/paper/2009.02439/) (2020)
- [Learning Neural Network Subspaces](https://mlsanity.com/paper/2102.10472/) (2021)
- [Large Scale Structure of Neural Network Loss Landscapes](https://mlsanity.com/paper/1906.04724/) (2019)
- [Weight-space symmetry in deep networks gives rise to permutation saddles, connected by equal-loss valleys across the loss landscape](https://mlsanity.com/paper/1907.02911/) (2019)
- [The jamming transition as a paradigm to understand the loss landscape of deep neural networks](https://mlsanity.com/paper/1809.09349/) (2018)
- [Shaping the learning landscape in neural networks around wide flat minima](https://mlsanity.com/paper/1905.07833/) (2019)

## Built on by (most cited first, 30 of 291)

- [Editing Models with Task Arithmetic](https://mlsanity.com/paper/2212.04089/) (2022)
- [TIES-Merging: Resolving Interference When Merging Models](https://mlsanity.com/paper/2306.01708/) (2023)
- [Transformers learn in-context by gradient descent](https://mlsanity.com/paper/2212.07677/) (2022)
- [Git Re-Basin: Merging Models modulo Permutation Symmetries](https://mlsanity.com/paper/2209.04836/) (2022)
- [Model Merging in LLMs, MLLMs, and Beyond: Methods, Theories, Applications and Opportunities](https://mlsanity.com/paper/2408.07666/) (2024)
- [Rewarded soups: towards Pareto-optimal alignment by interpolating weights fine-tuned on diverse rewards](https://mlsanity.com/paper/2306.04488/) (2023)
- [Arcee's MergeKit: A Toolkit for Merging Large Language Models](https://mlsanity.com/paper/2403.13257/) (2024)
- [Branch-Train-Merge: Embarrassingly Parallel Training of Expert Language Models](https://mlsanity.com/paper/2208.03306/) (2022)
- [Patching open-vocabulary models by interpolating weights](https://mlsanity.com/paper/2208.05592/) (2022)
- [ZipIt! Merging Models from Different Tasks without Training](https://mlsanity.com/paper/2305.03053/) (2023)
- [MiniCache: KV Cache Compression in Depth Dimension for Large Language Models](https://mlsanity.com/paper/2405.14366/) (2024)
- [Model merging with SVD to tie the Knots](https://mlsanity.com/paper/2410.19735/) (2024)
- [REPAIR: REnormalizing Permuted Activations for Interpolation Repair](https://mlsanity.com/paper/2211.08403/) (2022)
- [Fusing finetuned models for better pretraining](https://mlsanity.com/paper/2204.03044/) (2022)
- [Equivariant Architectures for Learning in Deep Weight Spaces](https://mlsanity.com/paper/2301.12780/) (2023)
- [Model Ratatouille: Recycling Diverse Models for Out-of-Distribution Generalization](https://mlsanity.com/paper/2212.10445/) (2022)
- [Deep Model Fusion: A Survey](https://mlsanity.com/paper/2309.15698/) (2023)
- [Soft Merging of Experts with Adaptive Routing](https://mlsanity.com/paper/2306.03745/) (2023)
- [Merging Multi-Task Models via Weight-Ensembling Mixture of Experts](https://mlsanity.com/paper/2402.00433/) (2024)
- [Permutation Equivariant Neural Functionals](https://mlsanity.com/paper/2302.14040/) (2023)
- [On the Symmetries of Deep Learning Models and their Internal Representations](https://mlsanity.com/paper/2205.14258/) (2022)
- [Deep Learning on Implicit Neural Representations of Shapes](https://mlsanity.com/paper/2302.05438/) (2023)
- [SD-LoRA: Scalable Decoupled Low-Rank Adaptation for Class Incremental Learning](https://mlsanity.com/paper/2501.13198/) (2025)
- [Re-basin via implicit Sinkhorn differentiation](https://mlsanity.com/paper/2212.12042/) (2022)
- [Analyzing and Reducing Catastrophic Forgetting in Parameter Efficient Tuning](https://mlsanity.com/paper/2402.18865/) (2024)
- [Merging LoRAs like Playing LEGO: Pushing the Modularity of LoRA to Extremes Through Rank-Wise Clustering](https://mlsanity.com/paper/2409.16167/) (2024)
- [Knowledge is a Region in Weight Space for Fine-tuned Language Models](https://mlsanity.com/paper/2302.04863/) (2023)
- [Revisiting Weighted Aggregation in Federated Learning with Neural Networks](https://mlsanity.com/paper/2302.10911/) (2023)
- [Graph Metanetworks for Processing Diverse Neural Architectures](https://mlsanity.com/paper/2312.04501/) (2023)
- [Linear Connectivity Reveals Generalization Strategies](https://mlsanity.com/paper/2205.12411/) (2022)

## Abstract (arXiv)

In this paper, we conjecture that if the permutation invariance of neural networks is taken into account, SGD solutions will likely have no barrier in the linear interpolation between them. Although it is a bold conjecture, we show how extensive empirical attempts fall short of refuting it. We further provide a preliminary theoretical result to support our conjecture. Our conjecture has implications for lottery ticket hypothesis, distributed training, and ensemble methods.