# Muon as a Residual Connection

arXiv: https://arxiv.org/abs/2607.01124 · HTML: https://mlsanity.com/paper/2607.01124/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 13 of its 14 references that are in the map.
- First posted on arXiv 1 Jul 2026.

## Builds on (most cited first, 13 of 13)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Identity Mappings in Deep Residual Networks](https://mlsanity.com/paper/1603.05027/) (2016)
- [Old Optimizer, New Norm: An Anthology](https://mlsanity.com/paper/2409.20325/) (2024)
- [Difference Target Propagation](https://mlsanity.com/paper/1412.7525/) (2014)
- [Understanding Gradient Orthogonalization for Deep Learning via Non-Euclidean Trust-Region Optimization](https://mlsanity.com/paper/2503.12645/) (2025)
- [Muon Outperforms Adam in Tail-End Associative Memory Learning](https://mlsanity.com/paper/2509.26030/) (2025)
- [Lions and Muons: Optimization via Stochastic Frank-Wolfe under Heavy-Tailed Noise](https://mlsanity.com/paper/2506.04192/) (2025)
- [How Muon's Spectral Design Benefits Generalization: A Study on Imbalanced Data](https://mlsanity.com/paper/2510.22980/) (2025)
- [Muon is Not That Special: Random or Inverted Spectra Work Just as Well](https://mlsanity.com/paper/2605.11181/) (2026)
- [Muon Dynamics as a Spectral Wasserstein Flow](https://mlsanity.com/paper/2604.04891/) (2026)
- [Why Muon Outperforms Adam: A Curvature Perspective](https://mlsanity.com/paper/2606.04662/) (2026)
- [Interpreting the Residual Stream of ResNet18](https://mlsanity.com/paper/2407.05340/) (2024)

## Abstract (arXiv)

Muon has recently emerged as one of the most effective optimizers for training large neural networks, yet its empirical success has been explained from several different perspectives. In this paper, we propose a simple mechanistic interpretation: Muon can be understood as an implicit residual connection during training. Specifically, orthogonalizing the update can sacrifice some immediate gradient fidelity while improving representation preservation for downstream layers. We study this trade-off in controlled linear optimization settings, where Muon can learn representations that are slower to fit a local target but easier for downstream layers to exploit. Our results suggest a conceptual explanation for Muon and a design perspective for optimizers that balance local descent with downstream usability.