# Relational Fisher Analysis: Dimensionality Reduction in Relational Data with Global Convergence +
## 5 Experiments
### Performance of RFA
#### 5.1.2 Comparison on Face Datasets  
Here, we tested RFA on the face recognition problems. The PIE and YaleB datasets were used. The details of these two datasets are shown in Table 5. For the corresponding experimental settings, we set the number of the nearest neighbors - \(k\) on the PIE dataset to 8 and \(k\) on the YaleB dataset to 18. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Dimension** & **LDA** & **MFA** & **PRPCA** & **RDA** & **RFA** \\ \hline
9 & 0.9108 \(\pm\) 0.0070 & 0.9570 \(\pm\) 0.0014 & 0.9230 \(\pm\) 0.0018 & 0.3699 \(\pm\) 0.0064 & **0.9580 \(\pm\) 0.0018** \\
13 & 0.9108 \(\pm\) 0.0070 & 0.9725 \(\pm\) 0.0022 & 0.9604 \(\pm\) 0.0028 & 0.3727 \(\pm\) 0.0065 & **0.9753 \(\pm\) 0.0022** \\
**15** & 0.9575 \(\pm\) 0.0023 & 0.9596 \(\pm\) 0.0018 & 0.9567 \(\pm\) 0.0022 & 0.3715 \(\pm\) 0.0087 & **0.9695 \(\pm\) 0.0011** \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Classification results obtained on the Letter dataset. Figure 8: Classification results obtained by RFA and MFA with different values of \(k_{p}\) on the USPS dataset. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Dimension** & **LDA** & **MFA** & **PRPCA** & **RDA** & **RFA** \\ \hline
9 & 0.9843 \(\pm\) 0.0020 & 0.9929 \(\pm\) 0.0013 & 0.9797 \(\pm\) 0.0027 & 0.9203 \(\pm\) 0.0046 & **0.9939 \(\pm\) 0.0006** \\
13 & - & 0.9940 \(\pm\) 0.0026 & 0.9604 \(\pm\) 0.0028 & 0.8856 \(\pm\) 0.0052 & **0.9950 \(\pm\) 0.0018** \\
15 & - & 0.9945 \(\pm\) 0.0018 & 0.9887 \(\pm\) 0.0008 & 0.9212 \(\pm\) 0.0050 & **0.9962 \(\pm\) 0.0013** \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Classification results obtained on the Pendigits dataset. Considering that RDA cannot perform well on multi-class classification problems, we used LPP instead as a compared method in these experiments. We used the 5-fold cross-validation to decide the value of parameter \(k\) for graph construction in LPP. We performed the experiments in different low-dimensional spaces on the PIE and YaleB datasets, and the experimental results are shown in Tables 6 and 7. We can see from the experimental results that RFA performs very well. Although LPP is an effective dimensionality reduction method for face recognition, RFA is significantly better than LPP. Moreover, RFA obtains comparable results with LDA and MFA. These results demonstrate the effectiveness of RFA in the face recognition applications. Additionally, convergence of RFA is verified on these two datasets. As illustrated in Figures 9 and 10, the value of \(\eta\) (trace ratio) decreases through the iterative procedures until it reaches the global optimal value \(\eta^{*}\) on both of the two datasets, which clearly shows the convergence of RFA. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Dimension** & **LDA** & **MFA** & **PRPCA** & **LPP** & **RFA** \\ \hline
9 & 0.9021 \(\pm\) 0.0070 & 0.9173 \(\pm\) 0.0045 & 0.7736 \(\pm\) 0.0113 & 0.8156 \(\pm\) 0.0089 & **0.9198 \(\pm\) 0.0026** \\
15 & 0.9430 \(\pm\) 0.0038 & 0.9526 \(\pm\) 0.0040 & 0.8810 \(\pm\) 0.0063 & 0.8930 \(\pm\) 0.0054 & **0.9540 \(\pm\) 0.0021** \\
20 & 0.9528 \(\pm\) 0.0031 & 0.9592 \(\pm\) 0.0027 & 0.9089 \(\pm\) 0.0062 & 0.9122 \(\pm\) 0.0044 & **0.9603 \(\pm\) 0.0030** \\
25 & 0.9597 \(\pm\) 0.0040 & 0.9617 \(\pm\) 0.0028 & 0.9224 \(\pm\) 0.0056 & 0.9262 \(\pm\) 0.0042 & **0.9621 \(\pm\) 0.0040** \\
30 & 0.9613 \(\pm\) 0.0037 & 0.9639 \(\pm\) 0.0042 & 0.9299 \(\pm\) 0.0041 & 0.9320 \(\pm\) 0.0031 & **0.9644 \(\pm\) 0.0027** \\
35 & 0.9623 \(\pm\) 0.0025 & 0.9643 \(\pm\) 0.0024 & 0.9342 \(\pm\) 0.0037 & 0.9352 \(\pm\) 0.0040 & **0.9660 \(\pm\) 0.0024** \\
40 & 0.9633 \(\pm\) 0.0019 & 0.9638 \(\pm\) 0.0021 & 0.9370 \(\pm\) 0.0049 & 0.9383 \(\pm\) 0.0036 & **0.9655 \(\pm\) 0.0021** \\
50 & 0.9639 \(\pm\) 0.0012 & 0.9643 \(\pm\) 0.0029 & 0.9394 \(\pm\) 0.0052 & 0.9411 \(\pm\) 0.0034 & **0.9656 \(\pm\) 0.0024** \\ C-1(67) & 0.9641 \(\pm\) 0.0048 & 0.9640 \(\pm\) 0.0034 & 0.9356 \(\pm\) 0.0045 & 0.9437 \(\pm\) 0.0035 & **0.9654 \(\pm\) 0.0037** \\ \hline \hline \end{tabular}
\end{table}