Rank-One Potential Geometry for Normalized Optimizers

Recent normalized optimizers such as Lion and Muon highlight the importance of geometry in modern optimizer design. We propose a unified framework that extends the Lion-K perspective to a broader class of normalized update rules by representing the momentum variable in an orthonormal rank-one system and defining an $\ell_1$-type coefficient potential, thereby covering SGD, Lion, and Muon within a single geometric view. We further study a regime with explicit time-dependent potentials, which is not covered by the static formulation, and show empirically that the resulting optimizer remains stable at ImageNet scale. On ViT-Base trained on ImageNet-1k, the proposed variant converges reliably and outperforms AdamW in our comparison, suggesting a route toward more systematic normalized-optimizer design.