proposition. Exact represented-matrix defect after one Adam step [ftip-00BE]
✍️source

    Let \(G_U=\nabla _UL(U_0,V_0)\), \(G_V=\nabla _VL(U_0,V_0)\), \(D_U=D_\epsilon (G_U)\), and \(D_V=D_\epsilon (G_V)\). Let \(W_1\) be the product after one Adam step from \((U_0,V_0)\) and \(\widetilde W_1\) the product after one step from \((U_0Q,V_0Q)\). Then

    \[ \begin {aligned} \widetilde W_1-W_1={}&-\eta \bigl (E_Q(G_U)V_0^{\mathsf T} +U_0E_Q(G_V)^{\mathsf T}\bigr )\\ &+\eta ^2\bigl (E_Q(G_U)D_V^{\mathsf T} +D_UE_Q(G_V)^{\mathsf T} +E_Q(G_U)E_Q(G_V)^{\mathsf T}\bigr ). \end {aligned} \]

    Proof. 

      Gauge-align the rotated factors back by \(Q^{\mathsf T}\). They are \(U_0-\eta (D_U+E_Q(G_U))\) and \(V_0-\eta (D_V+E_Q(G_V))\). Multiply, subtract \((U_0-\eta D_U)(V_0-\eta D_V)^{\mathsf T}\), and collect powers of \(\eta \).

      Proposition 4.4 and its proof are in Appendix B.5 of The loss does not see the basis, but Adam does[singh2026lossbasis].