\subsection{Proof of Proposition \ref{prop:opt2 and opt3}}
We rewrite $\widehat{L}$ and $\widehat{L}_0$ together with (\ref{opt-P2}) and (\ref{opt-P3}) below for the ease of readability.
\begin{equation}
\begin{aligned}
\widehat{L}:=\arg \min_{L} \quad  \widehat{R}_\mathcal{S}(L) \quad
\textrm{s.t.} \quad  \|L^\dagger L\|_{S_2}\leq \lambda_F    \\
\end{aligned} \tag{P2}
\end{equation}

\begin{equation}
\begin{aligned}
\widehat{L}_0:=\arg\min_{L} \quad & \widehat{R}_\mathcal{S}(L) \quad
\textrm{s.t.} \quad  \|\mathcal{P}_{\mathcal{S}_\mathcal{X}}^\dagger L^\dagger L\mathcal{P}_{\mathcal{S}_\mathcal{X}}\|_{S_2}\leq \lambda_F    \\
\end{aligned} \tag{P3}
\end{equation}
We have $\|\widehat{L}^\dagger\widehat{L}\|_{S_2}\leq \lambda_F$ by definition. From Lemma \ref{lem:norm inequality}, we also have 
\begin{eqnarray*}
   \|\mathcal{P}^\dagger_{\mathcal{S}_\mathcal{X}}\widehat{L}^\dagger \widehat{L}\mathcal{P}_{\mathcal{S}_\mathcal{X}}\|_{S_2} \leq  \|\widehat{L}^\dagger\widehat{L}\|_{S_2}.
\end{eqnarray*}
Thus, it holds that $\|\mathcal{P}^\dagger_{\mathcal{S}_\mathcal{X}}\widehat{L}^\dagger \widehat{L}\mathcal{P}_{\mathcal{S}_\mathcal{X}}\|_{S_2}\leq \lambda_F$.
Therefore, $\widehat{L}$ belongs to the feasible set of optimization problem (\ref{opt-P3}) and we can conclude that $\widehat{{R}}_\mathcal{S}(\widehat{L})$ is at least as small as $\widehat{{R}}_\mathcal{S}(\widehat{L}_0)$, i.e., 
\begin{eqnarray}
    \widehat{{R}}_\mathcal{S}(\widehat{L}_0) \leq \widehat{{R}}_\mathcal{S}(\widehat{L}). \label{emp of L0 smaller than L}
\end{eqnarray}
For the reverse inequality, note that $\widehat{L}_0 = \widehat{L}_0\mathcal{P}_{\mathcal{S}_\mathcal{X}}$. Therefore, $\|\mathcal{P}^\dagger_{\mathcal{S}_\mathcal{X}}\widehat{L}_0^\dagger \widehat{L}_0\mathcal{P}_{\mathcal{S}_\mathcal{X}}\|_{S_2}=\|\widehat{L}_0^\dagger \widehat{L}_0\|_{S_2}$ and $\widehat{L}_0$ belongs to the feasible set of (\ref{opt-P2}). Hence, 
\begin{eqnarray}
   \widehat{{R}}_\mathcal{S}(\widehat{L})\leq \widehat{{R}}_\mathcal{S}(\widehat{L}_0). \label{emp of L smaller than L0}
\end{eqnarray}
Based on (\ref{emp of L0 smaller than L}) and (\ref{emp of L smaller than L0}), we find that
\begin{eqnarray}
   \widehat{{R}}_\mathcal{S}(\widehat{L})= \widehat{{R}}_\mathcal{S}(\widehat{L}_0).
\end{eqnarray}
Furthermore, note that any ${L}$ within the feasible set of (\ref{opt-P2}) also belongs to the feasible set of (\ref{opt-P3}). Provided that ${L}\mathcal{P}_{\mathcal{S}_\mathcal{X}}=\widehat{L}_0$, we also conclude that $L$ is an optimal solution for (\ref{opt-P2}), since $\widehat{{R}}_\mathcal{S}({L}\mathcal{P}_{\mathcal{S}_\mathcal{X}})=\widehat{{R}}_\mathcal{S}(\widehat{L}_0)=\widehat{{R}}_\mathcal{S}(\widehat{L})$. 
