Definition. Gauge-equivariant optimizer [ftip-00B3]
Definition. Gauge-equivariant optimizer [ftip-00B3]
Let \(\mathcal A\) be a deterministic instance of the parameter-update rule of Definition [ftip-001I], acting on \((U,V,s)\). It is gauge-equivariant when for every \(Q\in \mathrm O(k)\) there is a state action \(\sigma _Q\) such that gauge-related initial states produce
\[ (\widetilde U_t,\widetilde V_t,\widetilde s_t) =(U_tQ,V_tQ,\sigma _Q(s_t)) \qquad (t\geq 0). \]This is Definition 3.1 of The loss does not see the basis, but Adam does[singh2026lossbasis]. Any randomness, schedules, stopping rule, and mixed update blocks must also be coupled as required by Remark [ftip-001J]; the definition does not hide them.