Directions That Don't Drift: Stiefel Manifold Routing for Transformer Attention

תקציר מקורי באנגליתarXiv:2609.19363v3 Announce Type: replace Abstract: The query and key projections $\WQ,\WK$ in attention are almost always trained by Euclidean optimizers with no geometric constraint. We constrain them to the Stiefel manifold and optimize with a Riemannian Adam carrying one scalar second moment per frame---the form of \citet{becigneul2019}, here extended to the compact, non-Hadamard $\St(d,r)$ with a tangent projector, step-norm cap, and polar retraction. Four propositions prove steepest descent in the embedded metric, gradient-scale independence, well-conditioning, and exact $\mathrm{O}(d)$-equivariance. A fifth records that weight decay has \emph{identically zero} Riemannian gradient on $\St(d,r)$ ($W{=}WI_r$ lies in the normal space), so decay cannot act on the constrained frames. On a