# Do We Really Need Permutations? Impact of Model Width on Linear Mode Connectivity

arXiv: https://arxiv.org/abs/2510.08023 · HTML: https://mlsanity.com/paper/2510.08023/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 37 of its 44 references that are in the map.
- First posted on arXiv 9 Oct 2025.

## Builds on (most cited first, 30 of 37)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Communication-Efficient Learning of Deep Networks from Decentralized Data](https://mlsanity.com/paper/1602.05629/) (2016)
- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [A Survey of Large Language Models](https://mlsanity.com/paper/2303.18223/) (2023)
- [The Lottery Ticket Hypothesis: Finding Sparse, Trainable Neural Networks](https://mlsanity.com/paper/1803.03635/) (2018)
- [WaveNet: A Generative Model for Raw Audio](https://mlsanity.com/paper/1609.03499/) (2016)
- [Model soups: averaging weights of multiple fine-tuned models improves accuracy without increasing inference time](https://mlsanity.com/paper/2203.05482/) (2022)
- [TIES-Merging: Resolving Interference When Merging Models](https://mlsanity.com/paper/2306.01708/) (2023)
- [Linear Mode Connectivity and the Lottery Ticket Hypothesis](https://mlsanity.com/paper/1912.05671/) (2019)
- [Merging Models with Fisher-Weighted Averaging](https://mlsanity.com/paper/2111.09832/) (2021)
- [Loss Surfaces, Mode Connectivity, and Fast Ensembling of DNNs](https://mlsanity.com/paper/1802.10026/) (2018)
- [Federated Optimization: Distributed Machine Learning for On-Device Intelligence](https://mlsanity.com/paper/1610.02527/) (2016)
- [Federated Learning with Matched Averaging](https://mlsanity.com/paper/2002.06440/) (2020)
- [Git Re-Basin: Merging Models modulo Permutation Symmetries](https://mlsanity.com/paper/2209.04836/) (2022)
- [Essentially No Barriers in Neural Network Energy Landscape](https://mlsanity.com/paper/1803.00885/) (2018)
- [The Role of Permutation Invariance in Linear Mode Connectivity of Neural Networks](https://mlsanity.com/paper/2110.06296/) (2021)
- [Sophia: A Scalable Stochastic Second-order Optimizer for Language Model Pre-training](https://mlsanity.com/paper/2305.14342/) (2023)
- [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)
- [REPAIR: REnormalizing Permuted Activations for Interpolation Repair](https://mlsanity.com/paper/2211.08403/) (2022)
- [Topology and Geometry of Half-Rectified Network Optimization](https://mlsanity.com/paper/1611.01540/) (2016)
- [Explaining Landscape Connectivity of Low-cost Solutions for Multilayer Nets](https://mlsanity.com/paper/1906.06247/) (2019)
- [Re-basin via implicit Sinkhorn differentiation](https://mlsanity.com/paper/2212.12042/) (2022)
- [Implicit Regularization Towards Rank Minimization in ReLU Networks](https://mlsanity.com/paper/2201.12760/) (2022)
- [On Connected Sublevel Sets in Deep Learning](https://mlsanity.com/paper/1901.07417/) (2019)
- [Going Beyond Linear Mode Connectivity: The Layerwise Linear Feature Connectivity](https://mlsanity.com/paper/2307.08286/) (2023)
- [On the loss landscape of a class of deep neural networks with no bad local valleys](https://mlsanity.com/paper/1809.10749/) (2018)
- [Landscape Connectivity and Dropout Stability of SGD Solutions for Over-parameterized Neural Networks](https://mlsanity.com/paper/1912.10095/) (2019)
- [Approaching Deep Learning through the Spectral Dynamics of Weights](https://mlsanity.com/paper/2408.11804/) (2024)

## Built on by (most cited first, 2 of 2)

- [Optimizer-Induced Mode Connectivity: From AdamW to Muon](https://mlsanity.com/paper/2605.09991/) (2026)
- [Beyond Structural Symmetries: Linear Mode Connectivity via Neuron Identifiability](https://mlsanity.com/paper/2606.04754/) (2026)

## Abstract (arXiv)

Recently, Ainsworth et al. empirically demonstrated that, given two independently trained models, applying a parameter permutation that preserves the input-output behavior allows the two models to be connected by a low-loss linear path. When such a path exists, the models are said to achieve linear mode connectivity (LMC). Prior studies, including Ainsworth et al.(2023), have reported that achieving LMC requires not only an appropriate permutation search but also sufficiently wide models (e.g., a 32 × width multiplier for ResNet-20). This is broadly believed to be because increasing the model width ensures a large enough space of candidate permutations, increasing the chance of finding one that yields LMC. In this work, we empirically demonstrate that, even without any permutations, simply widening the models is sufficient for achieving LMC when using a suitable softmax temperature calibration. We further explain why this phenomenon arises by analyzing intermediate layer outputs. Specifically, we introduce layerwise exponentially weighted connectivity (LEWC), which states that the output of each layer of the merged model can be represented as an exponentially weighted sum of the outputs of the corresponding layers of the original models. Consequently the merged model's output matches that of an ensemble of the original models, facilitating LMC. To the best of our knowledge, this work is the first to show that widening the model not only facilitates nonlinear mode connectivity, as suggested in prior research, but also significantly increases the possibility of achieving linear mode connectivity.