% !TEX root = ../main.tex
\subsection{Function-space Derivation of CVGP}
\label{assec:cvtgp_functionspace}
\input{appendix/cvtgp_derivation_f}

\clearpage
\subsection{Weight-space Derivation of CVGP}
\label{assec:cvtgp_weightspace}
\input{appendix/cvtgp_derivation_w}

\clearpage
\subsection{CVGP's lower-bound and its optimum}
\label{assec:cvgp_lower_bound_optimum}
\input{appendix/cvgp_lower_bound_optimum}

\clearpage
\subsection{Complexities of CVGP}
\label{assec:CVTGP_complexities}
CVGP maintains the time and space complexity of SVGP with less parameters.
This is because CVGP does not need to learn a free-form covariance matrix $\Sb$,
but only the coreset values $\XbC, \ybC$ and their weights $\betabC$, which is more efficient ---note that these three are $M$-dimensional vectors in the scalar case.
We describe the complexities of benchmarks and their parameters in Table~\ref{table:bigo2} below.

\begin{table}[h]
    \rowcolors{5}{}{gray!10}
    \begin{tabular}{*5c}
        \toprule
        & \multicolumn{4}{c}{Complexities$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$} \\
        \cmidrule(lr){2-4}
        Inference technique & Time  & Space & \# Parameter & Parameters\\    
        \midrule
        SparseGP \citep{titsias2009variational} &  $\;$$\;$$\;$$ \bigO{NM^2}$       &    $\;$$\;$$\;$$\bigO{NM^2}$    & $\bigO{M}$ & $\XbZ$ \\
        SVGP \citep{hensman2013gaussian} &     $\bigO{M^3}$   & $\bigO{M^2}$ &      $\;$$\;$$\bigO{M^2}$ & $\XbZ, \mb, \Sb$  \\
        CVGP &  $\bigO{M^3}$      & $\bigO{M^2}$ &  $\bigO{M}$ & $\XbC, \ybC, \betabC$ \\
        \bottomrule
    \end{tabular}
    \centering
    \caption{Computational analysis of CVGP and sparse $\gp$ alternatives:
    time and space complexities for obtaining an unbiased estimate of the log-marginal likelihood. CVGP enjoys same time and space complexity as SVGP, yet with a reduced variational parameter dimensionality. 
    Contrary to SVGP, CVGP does not learn a free-form covariance parameter $\Sb$, but only tempering-parameters $\betabC$ of same size as $\XbC, \ybC$.
    \label{table:bigo2}}
\end{table}
