Lemma. The gauge action preserves the product and loss [ftip-00AY]
✍️source

    For every \(A\in \mathrm {GL}(k)\),

    \[ W(UA,VA^{-\mathsf T})=UV^{\mathsf T} \quad \hbox {and}\quad L(UA,VA^{-\mathsf T})=L(U,V). \]

    Proof. 

      Since \((VA^{-\mathsf T})^{\mathsf T}=A^{-1}V^{\mathsf T}\), the first identity is \(UAA^{-1}V^{\mathsf T}=UV^{\mathsf T}\). Applying \(f\) gives the second.

      This is the factored-model calculation in Section 3 of The loss does not see the basis, but Adam does[singh2026lossbasis].