Definition. Objective-equivalent parameterizations [ftip-00BT]
Definition. Objective-equivalent parameterizations [ftip-00BT]
Two factor pairs \((U,V)\) and \((\widetilde U,\widetilde V)\) are objective-equivalent for \(L=f\circ W\) when
\[ W(U,V)=W(\widetilde U,\widetilde V). \]They then have the same scalar loss for every objective depending only on that represented matrix. Orthogonal gauge-related pairs are objective-equivalent by Lemma [ftip-00AY]; objective equivalence need not imply membership in the same orthogonal orbit.