
\textbf{Relating norms in $\mathcal{H}$ and $\mathbb{R}^n$:}
Above lines demonstrate how, for a given $L$, we may find a specific $\Mb$ that defines a metric on $\mathbb{R}^n$ which computes distances between points in $\mathcal{S}_\mathcal{X}\subset\mathcal{H}$ equally to $L$ using the KPCA basis for $\mathcal{S}_\mathcal{X}$.

We consider Schatten norm constraints on $L$ to rigorously define model classes for $L$ in (\ref{opt-P1}), (\ref{opt-P2}) and (\ref{opt-P3}). Hence, it is necessary to relate the Schatten norms of $L$ to Schatten norms of $\Mb$ so that constraints placed on $L$ in (\ref{opt-P3}) are comparable to those placed on $\Mb$ in $\mathbb{R}^n$. Following Lemma relate these norms.
\begin{lemma}\label{lem:relating norms}
    Let $\phi_1, \ldots, \phi_n$ be a set of features corresponding to a random set of triplets and $\mathcal{S}_\mathcal{X}$ is the span of feature points. For any $\phi_\xb\in \mathcal{S}_\mathcal{X}$ and $L: \mathcal{H}\rightarrow \mathcal{H}$, there exists a semidefinite matrix $\Mb \in \mathbb{R}^{n\times n}$ such that $\|L\phi_\xb\|_\mathcal{H}=\|\varphi_\xb\|_\Mb$ and $\|\mathcal{P}^\dagger_{\mathcal{S}_\mathcal{X}}L^\dagger L\mathcal{P}_{\mathcal{S}_\mathcal{X}}\|_{S_p}=\|\Mb\|_p$ $\forall p$.
\end{lemma}
Now, we can set up the finite dimensional optimization problem to learn a finite dimensional metric $\Mb$ that will enable us to find $\widehat{L}_0$.
