%%i%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Experiments}
\label{sec:exp}
%%i%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


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\subsection{Datasets clustering}
%%ii-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%

\begin{table}[!ht] 
\begin{center}
\begin{tabular}{|l r|}
\hline
Name & Shape $(n,d)$\\% & Num. clusters\\
\hline
yeast & $(1484, 9)$\\% &  10\\
faults & $(1941, 28)$\\% &  7\\
mfeat & $(2000, 77)$\\% &  10\\
cloud & $(2048, 11)$\\% &  8\\
segmentation & $(2310, 20)$\\%T &  7\\
spam & $(4601, 58)$\\% &  10\\
%% musk_clean& 6598 167 &  2\\
optdigits & $(5620, 65)$\\% &  10\\
landsat & $(6435, 37)$\\% &  6\\
pendigits& $(10992, 17)$\\% &  10\\
letter& $(20000, 17)$ \\%&  26
%%shuttle & $(58000, 10)$\\% &  7\\
%corel & $(68040, 26)$ &  16\\
\hline
\end{tabular}
\end{center}
\caption{{\bf The dataset $\datasetCLU$, consisting of 10 individual datasets.}
The shape of a dataset: $n$: number of observations/points; $d$: dimension.
Datasets selected in clustering experiments, see \cite{celebi2013comparative}.
} 
\label{tab:datasetCLU} 
\end{table} 

\begin{table}[!ht] 
\begin{center}
\begin{tabular}{| rr | rr | rr| rr |}
\hline
id & $n$ & id & $n$ & id & $n$ & id & $n$\\
\hline
cluster\_0 &     20 & cluster\_1 &       19 & cluster\_2 &       11 & cluster\_3 &       49\\
cluster\_4 &       45 & cluster\_5 &       77 & cluster\_6 &       69 & cluster\_7 &      101\\
cluster\_8 &       90 & cluster\_9 &      161 & cluster\_10 &      149 & cluster\_11 &      176 \\
cluster\_12 &      100 & cluster\_13 &       92 & cluster\_14 &      165 & cluster\_15 &      119 \\
\hline
\end{tabular}
\end{center}
\caption{{\bf The \datasetHMM dataset: 16 clusters in dimension $d=400$ -- from \cite{vicedomini2022profileview}.}
 We use the clusters as individual datasets.}
\label{tab:datasetHMM} 
\end{table} 

\clearpage

%%ii-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%
\subsubsection{Dataset \datasetCLU: comparison \algoexact vs \algobfgs}
%%ii-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%


\begin{figure}[!htb]
\begin{center}
\begin{tabular}{cc}
\includegraphics[width=.5\linewidth]{\wfig/DatasetCLU-dataset-SC-ratios-times-BFGS.png}&
\includegraphics[width=.5\linewidth]{\wfig/DatasetCLU-dataset-SC-ratios-values-BFGS.png}\\
Ratios $\timeexact / \timebfgs$ & Ratios $\Feta{\optexact} / \Feta{\optbfgs}$
\end{tabular}
\end{center}
\caption{{\bf \datasetCLU: \algoexact vs \algobfgs.}}
\label{fig:DatasetCLU-times-values-exact-BFGS}
\end{figure}

\input{DatasetCLU-dataset-SC-ratios-times-BFGS.tex}

\input{DatasetCLU-dataset-SC-ratios-values-BFGS.tex}
\FloatBarrier

\clearpage
\subsubsection{Dataset \datasetCLU: comparison \algoexact vs \algolbfgs}
%%ii-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%


\begin{figure}[htb]
\begin{center}
\begin{tabular}{cc}
\includegraphics[width=.5\linewidth]{\wfig/DatasetCLU-dataset-SC-ratios-times-LBFGS.png} &
\includegraphics[width=.5\linewidth]{\wfig/DatasetCLU-dataset-SC-ratios-values-LBFGS.png}\\
$\timeexact / \timelbfgs$ &  $\Feta{\optexact} / \Feta{\optlbfgs}$
\end{tabular}
\end{center}
\caption{{\bf \datasetCLU: \algoexact vs \algolbfgs.}}
\label{fig:DatasetCLU-times-values-exact-LBFGS}
\end{figure}

\input{DatasetCLU-dataset-SC-ratios-times-LBFGS.tex}

\input{DatasetCLU-dataset-SC-ratios-values-LBFGS.tex}
\FloatBarrier


\subsubsection{Illustration: \datasetCLU, dataset yeast\_std}

\begin{figure}[htb]
\begin{center}
\begin{tabular}{cc}
\includegraphics[width=0.404\textwidth]{\wfig/yeast_std-SC-values-R2-function-of-eta.png} & \includegraphics[width=0.404\textwidth]{\wfig/yeast_std-SC-stacked-barplot.png}\\
{\scriptsize Dataset yeast\_std: dual plot $\Feta$ and $R^2$} &Dataset yeast\_std: step types in trajectory\\
\includegraphics[width=0.404\textwidth]{\wfig/yeast_std-SC-time-ratios.png} & \includegraphics[width=0.404\textwidth]{\wfig/yeast_std-SC-mean-SESC-outlier-cost.png}\\
{\scriptsize Ratio $\timeexact / \timebfgs$} &Mean SESC outlier cost\\
\includegraphics[width=0.404\textwidth]{\wfig/yeast_std-SC-mean-COM-outlier-cost.png} & \includegraphics[width=0.404\textwidth]{\wfig/yeast_std-SC-outlier-num-ratio.png}\\
{\scriptsize Mean COM outlier cost} &Outliers: num(COM)/num(SESC)\\
\includegraphics[width=0.404\textwidth]{\wfig/yeast_std-SC-dist-comparisons.png}\\
{\scriptsize Distance between points}
\end{tabular}
\end{center}
\caption{{\bf \datasetCLU, dataset yeast\_std -- shape (1484,  9)}}
\label{fig:qq-yeast_std}
\end{figure}
\begin{figure}[htb]
\begin{center}
\begin{tabular}{cc}
%%
%\includegraphics[width=0.306\textwidth]{\wfig/yeast_std--sescmu1dot00-sesceta0dot20--randinit0-projection-plot.png} & 
%\includegraphics[width=0.306\textwidth]{\wfig/yeast_std--sescmu1dot00-sesceta0dot40--randinit0-projection-plot.png} & 
%\includegraphics[width=0.306\textwidth]{\wfig/yeast_std--sescmu1dot00-sesceta0dot60--randinit0-projection-plot.png}\\
%%
\includegraphics[width=0.45\textwidth]{\wfig/yeast_std--sescmu1dot00-sesceta0dot10--randinit0-projection-plot.png} & 
\includegraphics[width=0.45\textwidth]{\wfig/yeast_std--sescmu1dot00-sesceta0dot30--randinit0-projection-plot.png}\\
{\scriptsize Projection plot yeast\_std-, $\eta=0.1$} &Projection plot yeast\_std-, $\eta=0.3$\\
%%
\includegraphics[width=0.45\textwidth]{\wfig/yeast_std--sescmu1dot00-sesceta0dot50--randinit0-projection-plot.png} & 
\includegraphics[width=0.45\textwidth]{\wfig/yeast_std--sescmu1dot00-sesceta0dot70--randinit0-projection-plot.png}\\
{\scriptsize Projection plot yeast\_std-, $\eta=0.5$} &{\scriptsize Projection plot yeast\_std-, $\eta=0.7$}\\
%%
\includegraphics[width=0.45\textwidth]{\wfig/yeast_std--sescmu1dot00-sesceta0dot80--randinit0-projection-plot.png} & 
\includegraphics[width=0.45\textwidth]{\wfig/yeast_std--sescmu1dot00-sesceta0dot90--randinit0-projection-plot.png}\\
{\scriptsize Projection plot yeast\_std-, $\eta=0.8$} &{\scriptsize Projection plot yeast\_std-, $\eta=0.9$}
\end{tabular}
\end{center}
\caption{{\bf \datasetCLU, dataset yeast\_std -- shape (1484,  9)}}
\label{fig:pp-yeast_std}
\end{figure}
\FloatBarrier

\subsubsection{Illustration: \datasetCLU, dataset yeast\_std}

\begin{figure}[htb]
\begin{center}
\begin{tabular}{cc}
\includegraphics[width=0.404\textwidth]{\wfig/spam_std-SC-values-R2-function-of-eta.png} & \includegraphics[width=0.404\textwidth]{\wfig/spam_std-SC-stacked-barplot.png}\\
{\scriptsize Dataset spam\_std: dual plot $\Feta$ and $R^2$} &Dataset spam\_std: step types in trajectory\\
\includegraphics[width=0.404\textwidth]{\wfig/spam_std-SC-time-ratios.png} & \includegraphics[width=0.404\textwidth]{\wfig/spam_std-SC-mean-SESC-outlier-cost.png}\\
{\scriptsize Ratio $\timeexact / \timebfgs$} &Mean SESC outlier cost\\
\includegraphics[width=0.404\textwidth]{\wfig/spam_std-SC-mean-COM-outlier-cost.png} & \includegraphics[width=0.404\textwidth]{\wfig/spam_std-SC-outlier-num-ratio.png}\\
{\scriptsize Mean COM outlier cost} &Outliers: num(COM)/num(SESC)\\
\includegraphics[width=0.404\textwidth]{\wfig/spam_std-SC-dist-comparisons.png}\\
{\scriptsize Distance between points}
\end{tabular}
\end{center}
\caption{{\bf Dataset spam\_std -- shape (4601,  58)}}
\label{fig:qq-spam_std}
\end{figure}

\begin{figure}[htb]
\begin{center}
\begin{tabular}{cc}
%\includegraphics[width=0.45\textwidth]{\wfig/spam_std--sescmu1dot00-sesceta0dot20--randinit0-projection-plot.png} & 
%\includegraphics[width=0.45\textwidth]{\wfig/spam_std--sescmu1dot00-sesceta0dot40--randinit0-projection-plot.png} & 
%\includegraphics[width=0.45\textwidth]{\wfig/spam_std--sescmu1dot00-sesceta0dot60--randinit0-projection-plot.png}\\
\includegraphics[width=0.45\textwidth]{\wfig/spam_std--sescmu1dot00-sesceta0dot10--randinit0-projection-plot.png} & 
\includegraphics[width=0.45\textwidth]{\wfig/spam_std--sescmu1dot00-sesceta0dot30--randinit0-projection-plot.png}\\
{\scriptsize Projection plot spam\_std-, $\eta=0.1$} &Projection plot spam\_std-, $\eta=0.3$\\
%%
\includegraphics[width=0.45\textwidth]{\wfig/spam_std--sescmu1dot00-sesceta0dot50--randinit0-projection-plot.png} & 
\includegraphics[width=0.45\textwidth]{\wfig/spam_std--sescmu1dot00-sesceta0dot70--randinit0-projection-plot.png}\\
{\scriptsize Projection plot spam\_std-, $\eta=0.5$} &{\scriptsize Projection plot spam\_std-, $\eta=0.7$}\\
%%
\includegraphics[width=0.45\textwidth]{\wfig/spam_std--sescmu1dot00-sesceta0dot80--randinit0-projection-plot.png} & 
\includegraphics[width=0.45\textwidth]{\wfig/spam_std--sescmu1dot00-sesceta0dot90--randinit0-projection-plot.png}\\
{\scriptsize Projection plot spam\_std-, $\eta=0.8$} &{\scriptsize Projection plot spam\_std-, $\eta=0.9$}
\end{tabular}
\end{center}
\caption{{\bf Dataset spam\_std -- shape (4601,  58)}}
\label{fig:pp-spam_std}
\end{figure}
\FloatBarrier


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\clearpage
\subsection{Dataset \datasetHMM}
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\subsubsection{\datasetHMM: comparison \algoexact vs \algobfgs}
%%ii-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%

\begin{figure}[htb]
\begin{center}
\begin{tabular}{cc}
\includegraphics[width=.5\linewidth]{\wfig/ProtHMM-dataset-SC-ratios-times-BFGS.png}&
\includegraphics[width=.5\linewidth]{\wfig/ProtHMM-dataset-SC-ratios-values-BFGS.png}\\
%%
Ratios $\timeexact / \timebfgs$ & Ratios $\Feta{\optexact} / \Feta{\optbfgs}$
\end{tabular}
\end{center}
\caption{{\bf \datasetHMM: \algoexact vs \algobfgs.}}
\label{fig:ProtHMM-times-values-exact-BFGS}
\end{figure}

\clearpage
\input{ProtHMM-dataset-SC-ratios-times-BFGS.tex}

\input{ProtHMM-dataset-SC-ratios-values-BFGS.tex}
\FloatBarrier

\subsubsection{\datasetHMM: comparison \algoexact vs \algolbfgs}
%%ii-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%


\begin{figure}[htb]
\begin{center}
\begin{tabular}{cc}
\includegraphics[width=.5\linewidth]{\wfig/ProtHMM-dataset-SC-ratios-times-LBFGS.png}&
\includegraphics[width=.5\linewidth]{\wfig/ProtHMM-dataset-SC-ratios-values-LBFGS.png}\\
Ratios $\timeexact / \timelbfgs$ & Ratios $\Feta{\optexact} / \Feta{\optlbfgs}$
\end{tabular}
\end{center}
\caption{{\bf \datasetHMM: \algoexact vs \algolbfgs.}}
\label{fig:ProtHMM-times-values-exact-LBFGS}
\end{figure}


\clearpage

\input{ProtHMM-dataset-SC-ratios-times-LBFGS.tex}

\input{ProtHMM-dataset-SC-ratios-values-LBFGS.tex}
\FloatBarrier

\clearpage
\subsubsection{Illustration: \datasetHMM, Cluster 10}

\begin{figure}[htb]
\begin{center}
\begin{tabular}{cc}
\includegraphics[width=0.404\textwidth]{\wfig/cluster_10-SC-values-R2-function-of-eta.png} & \includegraphics[width=0.404\textwidth]{\wfig/cluster_10-SC-stacked-barplot.png}\\
{\scriptsize Dataset cluster\_10: dual plot $\Feta$ and $R^2$} &Dataset cluster\_10: step types in trajectory\\
\includegraphics[width=0.404\textwidth]{\wfig/cluster_10-SC-time-ratios.png} & \includegraphics[width=0.404\textwidth]{\wfig/cluster_10-SC-mean-SESC-outlier-cost.png}\\
{\scriptsize Ratio $\timeexact / \timebfgs$} &Mean SESC outlier cost\\
\includegraphics[width=0.404\textwidth]{\wfig/cluster_10-SC-mean-COM-outlier-cost.png} & \includegraphics[width=0.404\textwidth]{\wfig/cluster_10-SC-outlier-num-ratio.png}\\
{\scriptsize Mean COM outlier cost} &Outliers: num(COM)/num(SESC)\\
\includegraphics[width=0.404\textwidth]{\wfig/cluster_10-SC-dist-comparisons.png}\\
{\scriptsize Distance between points}
\end{tabular}
\end{center}
\caption{{\bf Dataset cluster\_10 -- shape (149,  400)}}
\label{fig:qq-cluster_10}
\end{figure}
\begin{figure}[htb]

\begin{center}
\begin{tabular}{cc}
%% \includegraphics[width=0.45\textwidth]{\wfig/cluster_10--sescmu1dot00-sesceta0dot20--randinit0-projection-plot.png} & 
%% \includegraphics[width=0.45\textwidth]{\wfig/cluster_10--sescmu1dot00-sesceta0dot40--randinit0-projection-plot.png} & 
%% \includegraphics[width=0.45\textwidth]{\wfig/cluster_10--sescmu1dot00-sesceta0dot60--randinit0-projection-plot.png}\\
%%
\includegraphics[width=0.45\textwidth]{\wfig/cluster_10--sescmu1dot00-sesceta0dot10--randinit0-projection-plot.png} & 
\includegraphics[width=0.45\textwidth]{\wfig/cluster_10--sescmu1dot00-sesceta0dot30--randinit0-projection-plot.png}\\
{\scriptsize Projection plot cluster\_10-, $\eta=0.1$} &{\scriptsize Projection plot cluster\_10-, $\eta=0.3$}\\
%%
\includegraphics[width=0.45\textwidth]{\wfig/cluster_10--sescmu1dot00-sesceta0dot50--randinit0-projection-plot.png} & 
\includegraphics[width=0.45\textwidth]{\wfig/cluster_10--sescmu1dot00-sesceta0dot70--randinit0-projection-plot.png}\\
{\scriptsize Projection plot cluster\_10-, $\eta=0.5$} &Projection plot cluster\_10-, $\eta=0.7$\\
%%
\includegraphics[width=0.45\textwidth]{\wfig/cluster_10--sescmu1dot00-sesceta0dot80--randinit0-projection-plot.png} & 
\includegraphics[width=0.45\textwidth]{\wfig/cluster_10--sescmu1dot00-sesceta0dot90--randinit0-projection-plot.png}\\
{\scriptsize Projection plot cluster\_10-, $\eta=0.8$} &{\scriptsize Projection plot cluster\_10-, $\eta=0.9$}
\end{tabular}
\end{center}
\caption{{\bf Dataset cluster\_10 -- shape (149,  400)}}
\label{fig:pp-cluster_10}
\end{figure}
\FloatBarrier



\clearpage

\paragraph{Illustration: \datasetHMM, Cluster 11.}

\begin{figure}[htb]
\begin{center}
\begin{tabular}{cc}
\includegraphics[width=0.404\textwidth]{\wfig/cluster_11-SC-values-R2-function-of-eta.png} & \includegraphics[width=0.404\textwidth]{\wfig/cluster_11-SC-stacked-barplot.png}\\
{\scriptsize Dataset cluster\_11: dual plot $\Feta$ and $R^2$} &Dataset cluster\_11: step types in trajectory\\
\includegraphics[width=0.404\textwidth]{\wfig/cluster_11-SC-time-ratios.png} & \includegraphics[width=0.404\textwidth]{\wfig/cluster_11-SC-mean-SESC-outlier-cost.png}\\
{\scriptsize Ratio $\timeexact / \timebfgs$} &Mean SESC outlier cost\\
\includegraphics[width=0.404\textwidth]{\wfig/cluster_11-SC-mean-COM-outlier-cost.png} & \includegraphics[width=0.404\textwidth]{\wfig/cluster_11-SC-outlier-num-ratio.png}\\
{\scriptsize Mean COM outlier cost} &Outliers: num(COM)/num(SESC)\\
\includegraphics[width=0.404\textwidth]{\wfig/cluster_11-SC-dist-comparisons.png}\\
{\scriptsize Distance between points}
\end{tabular}
\end{center}
\caption{{\bf Dataset cluster\_11 -- shape (176,  400)}}
\label{fig:qq-cluster_11}
\end{figure}
\begin{figure}[htb]

\begin{center}
\begin{tabular}{ccc}
%% \includegraphics[width=0.45\textwidth]{\wfig/cluster_11--sescmu1dot00-sesceta0dot20--randinit0-projection-plot.png} & 
%% \includegraphics[width=0.45\textwidth]{\wfig/cluster_11--sescmu1dot00-sesceta0dot40--randinit0-projection-plot.png} & 
%% \includegraphics[width=0.45\textwidth]{\wfig/cluster_11--sescmu1dot00-sesceta0dot60--randinit0-projection-plot.png}\\
%%
\includegraphics[width=0.45\textwidth]{\wfig/cluster_11--sescmu1dot00-sesceta0dot10--randinit0-projection-plot.png} & 
\includegraphics[width=0.45\textwidth]{\wfig/cluster_11--sescmu1dot00-sesceta0dot30--randinit0-projection-plot.png}\\
{\scriptsize Projection plot cluster\_11-, $\eta=0.1$} &{\scriptsize Projection plot cluster\_11-, $\eta=0.3$}\\
%%
\includegraphics[width=0.45\textwidth]{\wfig/cluster_11--sescmu1dot00-sesceta0dot50--randinit0-projection-plot.png} & 
\includegraphics[width=0.45\textwidth]{\wfig/cluster_11--sescmu1dot00-sesceta0dot70--randinit0-projection-plot.png}\\
{\scriptsize Projection plot cluster\_11-, $\eta=0.5$} &{\scriptsize Projection plot cluster\_11-, $\eta=0.7$}\\
%%
\includegraphics[width=0.45\textwidth]{\wfig/cluster_11--sescmu1dot00-sesceta0dot80--randinit0-projection-plot.png} & 
\includegraphics[width=0.45\textwidth]{\wfig/cluster_11--sescmu1dot00-sesceta0dot90--randinit0-projection-plot.png}\\
{\scriptsize Projection plot cluster\_11-, $\eta=0.8$} &{\scriptsize Projection plot cluster\_11-, $\eta=0.9$}
\end{tabular}
\end{center}
\caption{{\bf Dataset cluster\_11 -- shape (176,  400)}}
\label{fig:pp-cluster_11}
\end{figure}
\FloatBarrier



%%ii-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%
\subsubsection{Dataset Arcene}
%%ii-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%


\begin{figure}[!htb]
\begin{center}
\begin{tabular}{cc}
\includegraphics[width=.5\linewidth]{\wfig/DatasetHD-dataset-SC-ratios-times-LBFGS.png}&
\includegraphics[width=.5\linewidth]{\wfig/DatasetHD-dataset-SC-ratios-values-LBFGS.png}\\
Ratios $\timeexact / \timebfgs$ & Ratios $\Feta{\optexact} / \Feta{\optlbfgs}$
\end{tabular}
\end{center}
\caption{{\bf Arcene: \algoexact vs \algolbfgs.}}
\label{fig:DatasetHD-times-values-exact-LBFGS}
\end{figure}

\input{DatasetHD-dataset-SC-ratios-times-LBFGS.tex}

\input{DatasetHD-dataset-SC-ratios-values-LBFGS.tex}
\FloatBarrier
