# PolarAdamW: Disentangling Spectral Control and Schur Gauge-Equivariance in Matrix Optimisation

Source: [PolarAdamW: Disentangling Spectral Control and Schur Gauge-Equivariance in Matrix Optimisation](<https://www.sota2.com/research/paper/6a016297dbc836b87d82315c>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2026-05-12T05:00:04.827Z

Authors: Haozhou Zhang

Published: 2026-05-08T00:19:13.000Z

## Abstract

Muon's matrix-level update couples two distinct effects: spectral control via a polar map, and equivariance under orthogonal changes of multiplicity-space basis (Schur gauge-equivariance). We separate them with PolarAdamW, a controlled hybrid that preserves Muon's polar spectral-norm control but breaks the gauge-equivariance, since AdamW's coordinatewise preconditioner is basis-dependent. Algorithmically, PolarAdamW applies Muon's Newton-Schulz polar map to AdamW's preconditioned direction rather than to raw momentum, at per-iteration wall-time comparable to Muon. We prove that Muon's polar step is Schur gauge-equivariant on multiplicity matrices while AdamW's coordinatewise step is not.   On DeiT-Tiny trained from scratch on four independently sampled 100-class subsets of ImageNet-1k, where multiplicity-basis freedom is trivial, PolarAdamW outperforms Muon by +1.93 pp in test accuracy on average and AdamW by +9.5 pp; under the 300-epoch DeiT-style recipe, it remains ahead of Muon by +1.37 pp and AdamW by +5.80 pp on average. On SO(3)-equivariant 3D point-cloud regression, where multiplicity-basis freedom is non-trivial, the ordering reverses: Muon outperforms PolarAdamW at every audited capacity, and the gap widens with capacity. Both matrix-polar optimisers continue to outperform AdamW. This double dissociation separates spectral control from Schur gauge-equivariance: the first composes well with AdamW preconditioning on standard transformers, while the second becomes consequential when multiplicity-basis freedom is structurally non-trivial.

## Recorded benchmark results

A research result applies to its stated dataset, split, metric and evaluation configuration. Compare like-for-like settings, preserve the metric's unit and direction, and distinguish paper-reported results from independent reproduction. The best result recorded here is not a claim of exhaustive, current global coverage.

- [Image Classification on ImageNet-1k random 100-class subsets 1.0 (test)](<https://www.sota2.com/research/sota/image-classification-on-imagenet-1k-random-100-class-subsets-1-0-test.md>): Image Classification; ImageNet-1k random 100-class subsets 1.0 (test); Accuracy (L10 Test): 86.56.

- [SO(3)-equivariant point-cloud regression on SO(3) point-cloud (test)](<https://www.sota2.com/research/sota/so-3-equivariant-point-cloud-regression-on-so-3-point-cloud-test.md>): SO(3)-equivariant point-cloud regression; SO(3) point-cloud (test); Mean Test MSE: 0.0063.

## Original sources and implementation

[Original paper](<https://arxiv.org/abs/2605.07067>)

[PDF](<https://arxiv.org/pdf/2605.07067.pdf>)