# How the Optimizer Shapes Learned Solutions in Equivariant Neural Networks

arXiv: https://arxiv.org/abs/2605.27662 · HTML: https://mlsanity.com/paper/2605.27662/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 30 references that are in the map.
- First posted on arXiv 26 May 2026.

## Builds on (most cited first, 22 of 22)

- [PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation](https://mlsanity.com/paper/1612.00593/) (2016)
- [Dynamic Graph CNN for Learning on Point Clouds](https://mlsanity.com/paper/1801.07829/) (2018)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Visualizing the Loss Landscape of Neural Nets](https://mlsanity.com/paper/1712.09913/) (2017)
- [Strategies for Pre-training Graph Neural Networks](https://mlsanity.com/paper/1905.12265/) (2019)
- [Automatic chemical design using a data-driven continuous representation of molecules](https://mlsanity.com/paper/1610.02415/) (2016)
- [Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds](https://mlsanity.com/paper/1802.08219/) (2018)
- [Sharp Minima Can Generalize For Deep Nets](https://mlsanity.com/paper/1703.04933/) (2017)
- [Attention is Not All You Need: Pure Attention Loses Rank Doubly Exponentially with Depth](https://mlsanity.com/paper/2103.03404/) (2021)
- [Implicit Regularization in Deep Matrix Factorization](https://mlsanity.com/paper/1905.13655/) (2019)
- [An Investigation into Neural Net Optimization via Hessian Eigenvalue Density](https://mlsanity.com/paper/1901.10159/) (2019)
- [Long Range Graph Benchmark](https://mlsanity.com/paper/2206.08164/) (2022)
- [Signal Propagation in Transformers: Theoretical Perspectives and the Role of Rank Collapse](https://mlsanity.com/paper/2206.03126/) (2022)
- [Approximately Equivariant Networks for Imperfectly Symmetric Dynamics](https://mlsanity.com/paper/2201.11969/) (2022)
- [Benchmarking Robustness of 3D Point Cloud Recognition Against Common Corruptions](https://mlsanity.com/paper/2201.12296/) (2022)
- [Approximation-Generalization Trade-offs under (Approximate) Group Equivariance](https://mlsanity.com/paper/2305.17592/) (2023)
- [Improving Equivariant Model Training via Constraint Relaxation](https://mlsanity.com/paper/2408.13242/) (2024)
- [Optimizers Qualitatively Alter Solutions And We Should Leverage This](https://mlsanity.com/paper/2507.12224/) (2025)
- [The Price of Freedom: Exploring Expressivity and Runtime Tradeoffs in Equivariant Tensor Products](https://mlsanity.com/paper/2506.13523/) (2025)
- [Relaxed Equivariance via Multitask Learning](https://mlsanity.com/paper/2410.17878/) (2024)
- [Learning (Approximately) Equivariant Networks via Constrained Optimization](https://mlsanity.com/paper/2505.13631/) (2025)
- [A Tale of Two Symmetries: Exploring the Loss Landscape of Equivariant Models](https://mlsanity.com/paper/2506.02269/) (2025)

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

- [The Loss Does Not See the Basis, but Adam Does](https://mlsanity.com/paper/2608.05136/) (2026)
- [Equivariance Breaks the Learning Rate](https://mlsanity.com/paper/2609.08381/) (2026)

## Abstract (arXiv)

Equivariant neural networks encode geometric symmetries by construction, yet they are often difficult to optimize and can underperform less constrained architectures. A growing body of work addresses this through architectural modifications such as constraint relaxation or approximate equivariance, while the role of the optimizer remains comparatively underexplored. We study this direction by comparing Muon and Adam across several equivariant and geometric architectures under pointcloud and molecular learning settings. On ModelNet40, where the comparison is clearest, Muon consistently improves over Adam across all architectures considered. We then analyze the trained ModelNet40 checkpoints through Hessian estimates, loss surface visualizations, and spectral properties of learned weights and intermediate representations. The checkpoints reached by Muon have larger Hessian curvature summaries but more regular loss surfaces, and their learned weights and representations have higher stable and effective ranks. These observations suggest that the interaction between optimizer design and geometric inductive bias deserves further attention from the community.