$B_j$ and ${{\boldsymbol{G}}}(t)$. Recall that we would like to minimize $${\left\langle{{\boldsymbol{C}}}, {{\boldsymbol{C}}}\right\rangle}_{{HS}} 
= {\left\langle{\mathcal{P}}_{{{\boldsymbol{U}}}(t)} {{\boldsymbol{C}}}{\mathcal{P}}_{{{\boldsymbol{U}}}(t)}, {\mathcal{P}}_{{{\boldsymbol{U}}}(t)} {{\boldsymbol{C}}}{\mathcal{P}}_{{{\boldsymbol{U}}}(t)}\right\rangle}_{{HS}} 
+ {\langle{\mathcal{Q}}_{{{\boldsymbol{U}}}(t)} {{\boldsymbol{C}}}{\mathcal{Q}}_{{{\boldsymbol{U}}}(t)}, {\mathcal{Q}}_{{{\boldsymbol{U}}}(t)} {{\boldsymbol{C}}}{\mathcal{Q}}_{{{\boldsymbol{U}}}(t)}\rangle}_{{HS}}.$$ The second term is non-negative and we cannot minimize it further. As for the first term, by choosing $$\left\{
\begin{split}
M &= N, \\
A(y,t) &=  {{\boldsymbol{U}}}^{\dagger}(t) a({{\boldsymbol{U}}}(t) y, t), \\
B_j(y,t) &= {{\boldsymbol{U}}}^{\dagger}(t) b_j({{\boldsymbol{U}}}(t)y,t), \\
\end{split}\right.$$ one could easily verify that ${\mathcal{P}}_{{{\boldsymbol{U}}}(t)} {{\boldsymbol{C}}}{\mathcal{P}}_{{{\boldsymbol{U}}}(t)} = 0$. With such choice, $${\left\lVert{{\boldsymbol{C}}}\right\rVert}_{{HS}} = {\left\lVert\sum_{j=1}^{N} {\mathcal{Q}}_{{{\boldsymbol{U}}}(t)} {\mathbb{E}}_{\theta_t} \bigl[b_j({{\boldsymbol{U}}}(t)y,t) b_j^{\dagger}({{\boldsymbol{U}}}(t)y,t)\bigr] {\mathcal{Q}}_{{{\boldsymbol{U}}}(t)}\right\rVert}_{{HS}}.$$ Hence the above choice must be optimal (although it does not imply uniqueness). However, the choice of ${{\boldsymbol{G}}}$ is indeed unique under [[our restriction]{}]{}. One could straightforwardly verify that this solution satisfies the first two equations in the first order stationary condition ([*i.e.*]{}, [Eq. ]{}); that is to say, the above choice yields ${{\boldsymbol{U}}}^{\dagger}(t) {{\boldsymbol{C}}}{{\boldsymbol{U}}}(t) = 0$. Though we don’t have to use first two parts in [Eq. ]{} to derive low-rank dynamics, it is still nice to observe the consistency.

Thus we have proved the first and second part of this theorem. The third part follows easily from the first part.

Let us come back to the two restrictions we made at the beginning of this section.

-   The reason 