\subsection{Proof of Theorem \ref{thm:generalization_error_withbounded_Nuclear_norm}}

%Let $\mathcal{S}$ be the set of random observations over triplets with $\mathcal{X}$ as the set of objects used to generate triplets in $\mathcal{S}$, and $\Psi_1,\ldots,\Psi_n \in \mathcal{H}$ be KPCA directions for the span $\mathcal{S}_{\mathcal{X}}$. 

Recall that the only difference between the setting in Theorem \ref{thm:generalization_error_withbounded_Fro_norm} and the setting in Theorem \ref{thm:generalization_error_withbounded_Nuclear_norm} is the constraint set. We replace the constraints $\|\mathcal{P}^\dagger_{\mathcal{S}_\mathcal{X}}L^\dagger L\mathcal{P}_{\mathcal{S}_\mathcal{X}}\|_{S_2}\leq \lambda_F$ and $\|\Mb\|_F\leq \lambda_F$
with $\|\mathcal{P}^\dagger_{\mathcal{S}_\mathcal{X}}L^\dagger L\mathcal{P}_{\mathcal{S}_\mathcal{X}}\|_{S_1}\leq \lambda_*$ and $\|\Mb\|_*\leq \lambda_*$ respectively. We update definitions accordingly. Then, the proof follows the same steps  with the proof of Theorem \ref{thm:generalization_error_withbounded_Fro_norm} until (\ref{finite_kpca}), where we have
\begin{eqnarray}
R(\widehat{L}_{n_0})-R(L^*_n)
    &\leq& \frac{4\alpha}{|\mathcal{S}|} \mathbb{E}_{\mathcal{S}}\left[\mathbb{E}_{\epsilon |\mathcal{S}} \left[\sup_L \text{Tr}\left(\Mb^{\mathcal{S}_\mathcal{X}}\sum_{t\in \mathcal{S}}\epsilon_t \Kb_t\right)\right]\right] + \beta\sqrt{\frac{2\ln{2/\delta}}{|\mathcal{S}|}}
   \label{kpca bound_nuclear norm}
\end{eqnarray}
We focus on the expression inside the expectations and we can write    
\begin{eqnarray}
\sup_L \text{Tr}\left(\Mb^{\mathcal{S}_\mathcal{X}}\sum_{t\in \mathcal{S}}\epsilon_t \Kb_t\right)
    &\overset{a}{\leq}& \sup_L\|\Mb^{\mathcal{S}_\mathcal{X}}\|\left\|\sum_{t\in \mathcal{S}}\epsilon_t \Kb_t\right\| \nonumber
    \\
    %&\overset{b}{\leq}& \mathbb{E} \sup_L \left[\|\Gb\|_{\text{F}} \|(2\Psi_c-\Psi_a-\Psi_b)\Sigma (\Psi_b-\Psi_a)^T\|_{\text{F}}\right] \\
    &{\leq}&  \sup_L\|\Mb^{\mathcal{S}_\mathcal{X}}\|_*\left\|\sum_{t\in \mathcal{S}}\epsilon_t \Kb_t\right\| \nonumber
     \\
    &\overset{b}{\leq}& \lambda_*\|\sum_{t\in \mathcal{S}}\epsilon_t \Kb_t\|  \label{bounding_trace_nuclear norm}
\end{eqnarray}
  Here, $(a)$ is from Hölder's Ineqaulity for Schatten norms and we recall that $\|\Mb^{\mathcal{S}_\mathcal{X}}\|_{*}\leq\lambda_*$ for $(b)$. Inserting (\ref{bounding_trace_nuclear norm}) into the expectations in (\ref{kpca bound_nuclear norm}), we can write
\begin{eqnarray*}
    \mathbb{E}_{\mathcal{S}}\left[\mathbb{E}_{\epsilon |\mathcal{S}} \left[\sup_L \sum_{t\in \mathcal{S}}\epsilon_t(\|L\phi_h-L\phi_i\|_\mathcal{H}^2-\|L\phi_h-L\phi_j\|_\mathcal{H}^2)\right] \right]
    \leq
\lambda_*\mathbb{E}_{\mathcal{S}}\left[\mathbb{E}_{\epsilon |\mathcal{S}} \left[\left\|\sum_{t\in \mathcal{S}}\epsilon_t \Kb_t\right\|\right]\right].
\end{eqnarray*}
Then, we have
\begin{eqnarray}
\lambda_*\mathbb{E}_{\mathcal{S}}\left[\mathbb{E}_{\epsilon |\mathcal{S}}\left[\left\|\sum_{t\in \mathcal{S}}\epsilon_t \Kb_t\right\|\right]\right]
   &\overset{c}{\leq}&\lambda_*\mathbb{E}_{\mathcal{S}}\left[\sqrt{2\left\|\sum_{t=1}^{|\mathcal{S}|}\mathbb{E}_{\mathcal{\epsilon}}\left[\Kb_t^2\right]\right\|\log 3|\mathcal{S}|}+2\log 3|\mathcal{S}|\right] \nonumber
    \\
     &\overset{d}{\leq}& \lambda_*\mathbb{E}_{\mathcal{S}}\left[\sqrt{12B^4|\mathcal{S}|\log 3|\mathcal{S}| } +2\log 3|\mathcal{S}|\right]\nonumber
         \\
     &{=}& \lambda_*\left(\sqrt{12B^4|\mathcal{S}|\log 3|\mathcal{S}| } +2\log 3|\mathcal{S}|\right) \label{matrixBern}
\end{eqnarray}
where we apply a matrix Bernstein bound to get $(c)$ (see Theorem 6.6.1 in \cite{tropp2015introduction}) and $(d)$ follows from (\ref{bounding_fro_norm of Kt}). Lastly, from (\ref{kpca bound_nuclear norm}) and (\ref{matrixBern}), we have 
\begin{eqnarray*}
    R(\widehat{L}_{n_0})-R(L^*_n)\leq 4\alpha\lambda_*\left(B^2\sqrt{12\frac{\log 3|\mathcal{S}|}{|\mathcal{S}|} } +\frac{2\log 3|\mathcal{S}|}{|\mathcal{S}|}\right)+12\alpha B^2\lambda_*\sqrt{\frac{2\ln{2/\delta}}{|\mathcal{S}|}},
\end{eqnarray*}
which completes the proof of Theorem \ref{thm:generalization_error_withbounded_Nuclear_norm}.