# Optimizing Mode Connectivity via Neuron Alignment

arXiv: https://arxiv.org/abs/2009.02439 · HTML: https://mlsanity.com/paper/2009.02439/

## Key facts

- Cited by 86 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Sep 2020.

## Built on by (most cited first, 30 of 86)

- [TIES-Merging: Resolving Interference When Merging Models](https://mlsanity.com/paper/2306.01708/) (2023)
- [Git Re-Basin: Merging Models modulo Permutation Symmetries](https://mlsanity.com/paper/2209.04836/) (2022)
- [The Role of Permutation Invariance in Linear Mode Connectivity of Neural Networks](https://mlsanity.com/paper/2110.06296/) (2021)
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- [ZipIt! Merging Models from Different Tasks without Training](https://mlsanity.com/paper/2305.03053/) (2023)
- [Toward Transparent AI: A Survey on Interpreting the Inner Structures of Deep Neural Networks](https://mlsanity.com/paper/2207.13243/) (2022)
- [REPAIR: REnormalizing Permuted Activations for Interpolation Repair](https://mlsanity.com/paper/2211.08403/) (2022)
- [Equivariant Architectures for Learning in Deep Weight Spaces](https://mlsanity.com/paper/2301.12780/) (2023)
- [Deep Model Fusion: A Survey](https://mlsanity.com/paper/2309.15698/) (2023)
- [Merging Multi-Task Models via Weight-Ensembling Mixture of Experts](https://mlsanity.com/paper/2402.00433/) (2024)
- [Parameter Competition Balancing for Model Merging](https://mlsanity.com/paper/2410.02396/) (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)
- [Linear Connectivity Reveals Generalization Strategies](https://mlsanity.com/paper/2205.12411/) (2022)
- [What Matters for Model Merging at Scale?](https://mlsanity.com/paper/2410.03617/) (2024)
- [Neural Functional Transformers](https://mlsanity.com/paper/2305.13546/) (2023)
- [Going Beyond Linear Mode Connectivity: The Layerwise Linear Feature Connectivity](https://mlsanity.com/paper/2307.08286/) (2023)
- [Navigating the Safety Landscape: Measuring Risks in Finetuning Large Language Models](https://mlsanity.com/paper/2405.17374/) (2024)
- [Concrete Subspace Learning based Interference Elimination for Multi-task Model Fusion](https://mlsanity.com/paper/2312.06173/) (2023)
- [Transformer Fusion with Optimal Transport](https://mlsanity.com/paper/2310.05719/) (2023)
- [Merging by Matching Models in Task Parameter Subspaces](https://mlsanity.com/paper/2312.04339/) (2023)
- [Equivariant Deep Weight Space Alignment](https://mlsanity.com/paper/2310.13397/) (2023)
- [Training-Free Pretrained Model Merging](https://mlsanity.com/paper/2403.01753/) (2024)
- [Deep Networks on Toroids: Removing Symmetries Reveals the Structure of Flat Regions in the Landscape Geometry](https://mlsanity.com/paper/2202.03038/) (2022)
- [Sub-MoE: Efficient Mixture-of-Expert LLMs Compression via Subspace Expert Merging](https://mlsanity.com/paper/2506.23266/) (2025)
- [Deterministic Nonsmooth Nonconvex Optimization](https://mlsanity.com/paper/2302.08300/) (2023)
- [Proving Linear Mode Connectivity of Neural Networks via Optimal Transport](https://mlsanity.com/paper/2310.19103/) (2023)
- [Random initialisations performing above chance and how to find them](https://mlsanity.com/paper/2209.07509/) (2022)
- [PopulAtion Parameter Averaging (PAPA)](https://mlsanity.com/paper/2304.03094/) (2023)

## Abstract (arXiv)

The loss landscapes of deep neural networks are not well understood due to their high nonconvexity. Empirically, the local minima of these loss functions can be connected by a learned curve in model space, along which the loss remains nearly constant; a feature known as mode connectivity. Yet, current curve finding algorithms do not consider the influence of symmetry in the loss surface created by model weight permutations. We propose a more general framework to investigate the effect of symmetry on landscape connectivity by accounting for the weight permutations of the networks being connected. To approximate the optimal permutation, we introduce an inexpensive heuristic referred to as neuron alignment. Neuron alignment promotes similarity between the distribution of intermediate activations of models along the curve. We provide theoretical analysis establishing the benefit of alignment to mode connectivity based on this simple heuristic. We empirically verify that the permutation given by alignment is locally optimal via a proximal alternating minimization scheme. Empirically, optimizing the weight permutation is critical for efficiently learning a simple, planar, low-loss curve between networks that successfully generalizes. Our alignment method can significantly alleviate the recently identified robust loss barrier on the path connecting two adversarial robust models and find more robust and accurate models on the path.