\subsection{Proof of Proposition \ref{prop:operationalizing}}
From Lemma \ref{lem:relating norms}, we have $\|\mathcal{P}^\dagger_{\mathcal{S}_\mathcal{X}}L^\dagger L\mathcal{P}_{\mathcal{S}_\mathcal{X}}\|_{S_p}=\|\Mb\|_p$ $\forall p$. Similarly, from the fact that $\|L\phi_\xb\|_\mathcal{H}=\|\varphi_\xb\|_\Mb$ (see Lemma \ref{lem:relating norms}), we have $|\|L\phi_h-L\phi_i\|^2-\|L\phi_h-L\phi_j\|^2|=|\|\varphi_h-\varphi_i\|_\Mb^2-\|\varphi_h-\varphi_j\|_\Mb^2|$ within $\mathcal{S}_{\mathcal{X}}$. Then, from Proposition \ref{prop:representer theorem}, we conclude that (\ref{opt-P3}) and $(\ref{opt-P4})$ have the same optimal value. Therefore, we have that
\begin{equation}
\begin{aligned}
\min_{L} \quad & \widehat{R}_\mathcal{S}(L)\\
\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}
is equal to
\begin{equation}
\begin{aligned}
\min_{\Mb} \quad & \widehat{\overline{R}}_\mathcal{S}(\Mb)\\
\textrm{s.t.} \quad &   \|\Mb\|_F \leq \lambda_F    \\
  &  \Mb \succeq 0,    \\
\end{aligned}\tag{P4}
\end{equation}
By definition, $\widehat{\Lb}_0$ is an optimal solution for (\ref{opt-P3}) and $\widehat{\Mb}$ is the optimal solution for (\ref{opt-P4}). Recall that there exists a psd matrix $\Mb$ for each pair of $L$ and $\mathcal{S}_\mathcal{X}$ from Lemma \ref{lem:relating norms}.



For the construction of $\widehat{L}_0$ from $\widehat{\Mb}$, we follow similar lines with the proof of Lemma \ref{lem:relating norms}.  $\widehat{L}_0$ is defined reversing equalities in Section \ref{sec:proof of lem relating norms} for $\Mb = \widehat{\Mb}$, from (\ref{firstline Lem1}) to (\ref{lastline lem1}). Therefore, we observe that $\widehat{\Mb}$ is the corresponding psd matrix for the pair $(\widehat{L}_0, \mathcal{S}_\mathcal{X})$. This is actually true for any $L$ provided that $L\mathcal{P}_{\mathcal{S}_\mathcal{X}}=\widehat{L}_0$.



