evaluation.tex

\subsection{Evaluation}
\begin{frame}{Baseline Systems}
    \textbf{Preprocessing:} Scaling, shifting and linear interpolation\\
    \textbf{Features:} Coordinates of 80 points (4 strokes with 20 points each)\\
    \textbf{Learning:} MLP, 1000 epochs, LR $\eta=0.1$, Momentum $\alpha=0.1$
\begin{table}[h]
    \centering
    \begin{tabular}{clrrr}
    \toprule
    \multirow{2}{*}{System}  & \multirow{2}{*}{Topology} & \multicolumn{3}{c}{Classification error}\\
          &                         & TOP1                   & TOP3                  & MER \\\midrule
    $B_1$ & 160:500:369             & $\SI{23.34}{\percent}$ & $\SI{6.80}{\percent}$ & $\SI{6.64}{\percent}$ \\
    $B_2$ & 160:500:500:369         & \underline{$\SI{21.51}{\percent}$} & $\SI{5.75}{\percent}$ & $\SI{5.67}{\percent}$ \\
    $B_3$ & 160:500:500:500:369     & $\SI{21.93}{\percent}$ & \underline{$\SI{5.74}{\percent}$} & \underline{$\SI{5.64}{\percent}$} \\
    % $B_4$ & 160:500:500:500:500:369 & $\SI{23.88}{\percent}$ & $\SI{6.12}{\percent}$ & $\SI{6.04}{\percent}$ \\
    \bottomrule
    \end{tabular}
    \caption{Baseline systems with three different classification error measures.
             All errors were measured on the test set.}
\label{table:baseline-systems}
\end{table}
\end{frame}

\subsection{Merged Symbols}
\begin{frame}[fragile]{Merged symbols (MER error)}
\begin{table}[ht]
    \centering
    \begin{tabular}{lc|lc}
        \toprule
        \multicolumn{2}{c}{Base symbol} & \multicolumn{2}{c}{equivalent symbols}\\
        \LaTeX         & Rendered       & \LaTeX                 & Rendered  \\\midrule
        \verb+\sum+    & $\sum$         & \verb+$\Sigma$+        & $\Sigma$\\
        \verb+\prod+   & $\prod$        & \verb+$\Pi$+           & $\Pi$\\
        ~              & ~              & \verb+$\sqcap$+        & $\sqcap$\\
        \verb+\coprod+ & $\coprod$      & \verb+$\amalg$+        & $\amalg$\\
        ~              & ~              & \verb+$\sqcup$+        & $\sqcup$\\
        \verb+\perp+   & $\perp$        & \verb+$\bot$+          & $\bot$\\
        \verb+\models+ & $\models$      & \verb+$\vDash$+        & $\vDash$\\
        \verb+|+       & |              & \verb+\mid+            & $\mid$  \\
        \verb+\Delta+  & $\Delta$       & \verb+$\triangle$+     & $\triangle$\\
        ~              & ~              & \verb+$\vartriangle$+  & $\vartriangle$\\
        \bottomrule
    \end{tabular}
\end{table}
\end{frame}

\begin{frame}[fragile]{Merged symbols (MER error)}
\begin{table}[ht]
    \centering
    \begin{tabular}{lc|lc}
        \toprule
        \multicolumn{2}{c}{Base symbol} & \multicolumn{2}{c}{equivalent symbols}\\
        \LaTeX         & Rendered       & \LaTeX                 & Rendered  \\\midrule
        \verb+\|+      & $\|$           & \verb+$\parallel$+     & $\parallel$\\
        \verb+\ohm+    & $\Omega$         & \verb+$\Omega$+      & $\Omega$\\
        \verb+\setminus+ & $\setminus$  & \verb+$\backslash$+    & $\backslash$\\
        \verb+\checked+ & {\mbox {\wasyfamily \char 8}} & \verb+$\checkmark$+    & $\checkmark$\\
        \verb+\&+      & $\&$           & \verb+$\with$+         & $\&$\\
        \verb+\#+      & $\#$           & \verb+$\sharp$+        & $\sharp$\\
        \verb+\S+      & $\S$           & \verb+$\mathsection$+  & $\mathsection$\\
        \verb+\nabla+  & $\nabla$       & \verb+\triangledown+   & $\triangledown$\\
        \verb+\lhd+    & $\lhd$         & \verb+$\triangleleft$+ & $\triangleleft$\\
        ~              & ~              & \verb+$\vartriangleleft$+ & $\vartriangleleft$\\
        \verb+\oiint+  & $\oiint$       & \verb+$\varoiint$+     & $\varoiint$\\
        \bottomrule
    \end{tabular}
\end{table}
\end{frame}


\begin{frame}[fragile]{Merged symbols (MER error)}
\begin{table}[ht]
    \centering
    \begin{tabular}{lc|lc}
        \toprule
        \multicolumn{2}{c}{Base symbol} & \multicolumn{2}{c}{equivalent symbols}\\
        \LaTeX         & Rendered       & \LaTeX                 & Rendered  \\\midrule
        \verb+\mathbb{R}+ & $\mathbb{R}$ & \verb+$\mathds{R}$+   & $\mathds{R}$\\
        \verb+\mathbb{Q}+ & $\mathbb{Q}$ & \verb+\mathds{Q}+     & $\mathds{Q}$\\
        \verb+\mathbb{Z}+ & $\mathbb{Z}$ & \verb+\mathds{Z}+     & $\mathds{Z}$\\
        \verb+\mathcal{A}+ & $\mathcal{A}$ & \verb+\mathscr{A}+  & $\mathscr{A}$\\
        \verb+\mathcal{D}+ & $\mathcal{D}$ & \verb+\mathscr{D}+  & $\mathscr{D}$\\
        \verb+\mathcal{N}+ & $\mathcal{N}$ & \verb+\mathscr{N}+  & $\mathscr{N}$\\
        \verb+\mathcal{R}+ & $\mathcal{R}$ & \verb+\mathscr{R}+  & $\mathscr{R}$\\
        \verb+\propto+ & $\propto$      & \verb+$\varpropto$+    & $\varpropto$\\
        \bottomrule
    \end{tabular}
\end{table}
\end{frame}

\subsection{Evaluation}
\begin{frame}{Preprocessing: Connect strokes}
\begin{table}[h]
    \centering
    \begin{tabular}{lrrrrrr}
    \toprule
    \multirow{2}{*}{System} & \multicolumn{6}{c}{Classification error}\\
    \cmidrule(l){2-7}
                                         & TOP1                   & change                 & TOP3                   & change                 & MER                & change \\\midrule
    $B_{1,\theta = \SI{5}{\pixel}}$ & $\SI{23.27}{\percent}$ & $\SI{-0.07}{\percent}$ &  $\SI{6.50}{\percent}$ & $\SI{-0.30}{\percent}$ & $\SI{6.37}{\percent}$ & $\SI{-0.27}{\percent}$ \\
    $B_{2,\theta = \SI{5}{\pixel}}$ & $\SI{21.20}{\percent}$ & $\SI{-0.31}{\percent}$ &  $\SI{5.59}{\percent}$ & $\SI{-0.16}{\percent}$ & $\SI{5.50}{\percent}$ & $\SI{-0.17}{\percent}$ \\
    $B_{3,\theta = \SI{5}{\pixel}}$ & $\SI{21.80}{\percent}$ & $\SI{-0.13}{\percent}$ &  $\SI{5.54}{\percent}$ & $\SI{-0.20}{\percent}$ & $\SI{5.47}{\percent}$ & $\SI{-0.17}{\percent}$ \\\midrule
    $B_{1,\theta = \SI{10}{\pixel}}$& $\SI{23.17}{\percent}$ & $\SI{-0.17}{\percent}$ &  $\SI{6.61}{\percent}$ & $\SI{-0.19}{\percent}$ & $\SI{6.47}{\percent}$ & $\SI{-0.17}{\percent}$ \\
    $B_{2,\theta = \SI{10}{\pixel}}$& \underline{$\SI{20.97}{\percent}$} & $\SI{-0.54}{\percent}$ &  $\SI{5.43}{\percent}$ & $\SI{-0.32}{\percent}$ & $\SI{5.34}{\percent}$ & $\SI{-0.33}{\percent}$ \\
    $B_{3,\theta = \SI{10}{\pixel}}$& $\SI{21.34}{\percent}$ & $\SI{-0.59}{\percent}$ & \underline{$\SI{5.42}{\percent}$} & $\SI{-0.32}{\percent}$ & \underline{$\SI{5.33}{\percent}$} & $\SI{-0.31}{\percent}$ \\\midrule
    $B_{1,\theta = \SI{20}{\pixel}}$& $\SI{22.81}{\percent}$ & $\SI{-0.53}{\percent}$ &  $\SI{6.28}{\percent}$ & $\SI{-0.52}{\percent}$ & $\SI{6.19}{\percent}$ & $\SI{-0.45}{\percent}$ \\
    $B_{2,\theta = \SI{20}{\pixel}}$& $\SI{21.61}{\percent}$ & $\SI{+0.10}{\percent}$ &  $\SI{5.79}{\percent}$ & $\SI{+0.04}{\percent}$ & $\SI{5.69}{\percent}$ & $\SI{+0.02}{\percent}$ \\
    $B_{3,\theta = \SI{20}{\pixel}}$& $\SI{21.71}{\percent}$ & $\SI{-0.22}{\percent}$ &  $\SI{5.55}{\percent}$ & $\SI{-0.19}{\percent}$ & $\SI{5.45}{\percent}$ & $\SI{-0.19}{\percent}$ \\
    \bottomrule
    \end{tabular}
    \caption{Models $B_1$--$B_4$ with additionally applied stroke
             connect algorithm.}
\label{table:stroke-connect-evaluation}
\end{table}
\end{frame}

\begin{frame}{Learning: Supervised layer-wise pretraining}
\begin{table}[tb]
    \centering
    \begin{tabular}{lrrrrrr}
    \toprule
    \multirow{2}{*}{System}& \multicolumn{6}{c}{Classification error}\\
    \cmidrule(l){2-7}
              & TOP1                   & change                 & TOP3                  & change                 & MER                & change \\\midrule
    $B_1$     & $\SI{23.34}{\percent}$ &                        & $\SI{6.80}{\percent}$ &                        & $\SI{6.64}{\percent}$ & \\%chktex 8
    $B_{2,p}$ & $\SI{19.89}{\percent}$ & $\SI{-1.62}{\percent}$ & $\SI{4.76}{\percent}$ & $\SI{-0.99}{\percent}$ & $\SI{4.68}{\percent}$ & $\SI{-0.99}{\percent}$\\
    $B_{3,p}$ & \underline{$\SI{19.43}{\percent}$} & $\SI{-2.50}{\percent}$ & \underline{$\SI{4.64}{\percent}$} & $\SI{-1.10}{\percent}$ & \underline{$\SI{4.54}{\percent}$} & $\SI{-1.10}{\percent}$\\
    % $B_{4,p}$ (pretrained) & $\SI{19.63}{\percent}$ & $\SI{-4.25}{\percent}$ & $\SI{4.66}{\percent}$ & $\SI{-1.46}{\percent}$ & $\SI{4.55}{\percent}$ & $\SI{-1.49}{\percent}$\\
    \bottomrule
    \end{tabular}
    \caption{Supervised layer-wise pretraining, 1000 epochs per layer}
\label{table:pretraining-copy-before}
\end{table}
\end{frame}

\definecolor{light-gray}{gray}{0.70}

\subsection{Optimized classifier}
\begin{frame}{Optimized classifier}
    \textbf{Preprocessing:} Connect strokes{\color{light-gray}, scale, shift and linear interpolation}\\
    \textbf{Features:} {\color{light-gray}Coordinates of 80 points (4 strokes with 20 points each),} re-curvature per stroke, ink, stroke count, aspect ratio\\
    \textbf{Learning:} {\color{light-gray}MLP, 1000 epochs, LR $\eta=0.1$, Momentum $\alpha=0.1$,} supervised layer-wise pretraining
\begin{table}[htb]
    \centering
    \begin{tabular}{lrrrrrr}
    \toprule
    \multirow{2}{*}{System}  & \multicolumn{6}{c}{Classification error}\\
    \cmidrule(l){2-7}
              & TOP1                   & change                 & TOP3                  & change                 & MER                   & change \\\midrule
    $B_{1,c}$ & $\SI{20.96}{\percent}$ & $\SI{-2.38}{\percent}$ & $\SI{5.24}{\percent}$ & $\SI{-1.56}{\percent}$ & $\SI{5.13}{\percent}$ & $\SI{-1.51}{\percent}$ \\
    $B_{2,c}$ & $\SI{18.26}{\percent}$ & $\SI{-3.25}{\percent}$ & $\SI{4.07}{\percent}$ & $\SI{-1.68}{\percent}$ & \underline{$\SI{3.98}{\percent}$} & $\SI{-1.69}{\percent}$ \\
    $B_{3,c}$ & \underline{$\SI{18.19}{\percent}$} & $\SI{-3.74}{\percent}$ & \underline{$\SI{4.06}{\percent}$} & $\SI{-1.68}{\percent}$ & $\SI{3.99}{\percent}$ & $\SI{-1.65}{\percent}$ \\
    % $B_{4,c}$ & $\SI{18.57}{\percent}$ & $\SI{-5.31}{\percent}$ & $\SI{4.25}{\percent}$ & $\SI{-1.87}{\percent}$ & $\SI{4.18}{\percent}$ & $\SI{-1.86}{\percent}$ \\
    \bottomrule
    \end{tabular}
    \caption{Error rates of the complex recognizer systems.}
\label{table:complex-recognizer-systems-evaluation}
\end{table}
\end{frame}