## 4 Experiments
### Parameter Study
#### 4.3.2 Sensitivity Analysis of Cluster Centers Initialization of the SPVEA  
In this subsection, we study the robustness of cluster center initialization procedure of the SPVEA. We apply SPVEA 30 times with different sets of initial guesses to each of the four face databases with \(r=1\) and obtain projection matrices. We measure the difference among these projection matrices by the following formula  
\[Diff(D_{V})=std(\left\|V(i)^{T}V(j)\right\|_{1}/D_{V}), \tag{22}\]  
where \(V(i)\) is the projection matrices obtained by SPVEA with ith set of initial guesses and \(D_{v}\) is the number of projection vectors of the projection matrix. \(Diff(D_{V})\) is to compute the standard deviation  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{\(p\)} & \multirow{2}{*}{**Database**} & \multicolumn{4}{c}{\(r\)} \\ \cline{3-6}  & & **1** & **2** & **3** & **4** \\ \hline \multirow{4}{*}{95} & JAFFE & 1.000 & 0.995 & 0.993 & 0.988 \\  & Yale & 0.643 & 0.737 & 0.806 & 0.811 \\  & AR (Normal + Lighting) & 0.568 & 0.663 & 0.753 & 0.787 \\  & AR (Normal + Sunglasses) & 0.604 & 0.683 & 0.751 & 0.784 \\  & AR (Normal + Scurves) & 0.448 & 0.600 & 0.693 & 0.742 \\ \hline \multirow{4}{*}{90} & JAFFE & 1.000 & 0.995 & 0.993 & 0.988 \\  & Yale & 0.646 & 0.744 & 0.802 & 0.810 \\  & AR (Normal + Lighting) & 0.556 & 0.657 & 0.749 & 0.787 \\  & AR (Normal + Sunglasses) & 0.596 & 0.683 & 0.760 & 0.787 \\  & AR (Normal + Scurves) & 0.472 & 0.600 & 0.702 & 0.742 \\ \hline \multirow{4}{*}{85} & JAFFE & 1.000 & 0.995 & 0.997 & 0.990 \\  & Yale & 0.654 & 0.739 & 0.796 & 0.805 \\  & AR (Normal + Lighting) & 0.588 & 0.709 & 0.753 & 0.786 \\  & AR (Normal + Sunglasses) & 0.652 & 0.714 & 0.782 & 0.800 \\  & AR (Normal + Scurves) & 0.476 & 0.603 & 0.722 & 0.744 \\ \hline \multirow{4}{*}{80} & JAFFE & 1.000 & 0.995 & 0.997 & 0.99 \\  & Yale & 0.681 & 0.729 & 0.794 & 0.816 \\  & AR (Normal + Lighting) & 0.604 & 0.726 & 0.747 & 0.789 \\  & AR (Normal + Sunglasses) & 0.820 & 0.803 & 0.791 & 0.820 \\  & AR (Normal + Scurves) & 0.504 & 0.620 & 0.716 & 0.747 \\ \hline \multirow{4}{*}{75} & JAFFE & 1.000 & 0.990 & 0.990 & 0.995 \\  & Yale & 0.681 & 0.763 & 0.809 & 0.816 \\ \cline{1-1}  & AR (Normal + Lighting) & 0.680 & 0.726 & 0.773 & 0.784 \\ \cline{1-1}  & AR (Normal + Sunglasses) & 0.816 & 0.840 & 0.869 & 0.815 \\ \cline{1-1}  & AR (Normal + Scurves) & 0.512 & 0.629 & 0.713 & 0.762 \\ \hline \hline \end{tabular}
\end{table}
Table 8: Classification Performance (averaged maximum classification rate (AMCR)) of TC-PCA for Various Parameters.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Database**} & \multicolumn{4}{c}{\(r\)} \\ \cline{2-5}  & **1** & **2** & **3** & **4** \\ \hline JAFFE & 1.000 & 0.975 & 0.987 & 0.995 \\ Yale & 0.659 & 0.744 & 0.809 & 0.812 \\ AR (Normal + Lighting) & 0.632 & 0.683 & 0.773 & 0.795 \\ AR (Normal + Sunglasses) & 0.812 & 0.860 & 0.829 & 0.802 \\ AR (Normal + Scurves) & 0.480 & 0.617 & 0.709 & 0.767 \\ \hline \hline \end{tabular}
\end{table}
Table 9: Classification Performance (AMCR) of TC-PCA with Holdout Estimation for the Trimming Parameter.  
of all the dot products of any two projection matrices obtained by different sets of initial guesses. If all projection matrices \(V(i)\) are similar to each other, the \(\|V(i)^{T}V(j)\|_{1}\) will be close to one and thus, obtain a very small standard deviation. Table 10 below shows the values of \(Diff(D_{V})\) to the four face databases under three settings: \(D_{V}=1,\ D_{V}=10\) and \(D_{V}=20\). We can see that all values are very small and close to zero. This shows that the projection vectors generated by SPVEA is robust to cluster center initialization.