# Relational Fisher Analysis: Dimensionality Reduction in Relational Data with Global Convergence +
## 5 Experiments
### Performance of RFA
#### 5.1.1 Comparison on Hand-Writing Datasets  
We first tested the performance of RFA on the hand-writing datasets. For clarity, the details of the used datasets are shown in Table 1.  
**[PERSON] dataset:** This dataset [55] is an ancient manuscript dataset, one image of which is shown in Figure 1, and we tried to identify the Arabic subwords on this dataset. In the experiment, we used a 50-page manuscript as the training set and 10 pages as the test set. We extracted the square-root velocity (SRV) representation [56] of the Arabic subwords. Then, we removed the outlier classes, including the classes that had less than 10 samples. Finally, we obtained a 174-class Arabic subword dataset with 17,543 samples for training and 3125 samples for test. In this experiment, we set the number of the nearest neighbors - \(k\) of the RFA to 8 and compared RFA with the LDA and MFA dimensionality reduction algorithms. The classification accuracies after using these three dimensionality reduction methods to map the data to different dimensionalities are shown in Figure 2. We can see that RFA is far better than the LDA algorithm and slightly better than the MFA algorithm. At the same time, when the dimensionality is from 50 to \(C-1\) (\(C\) is the number of classes), the correct rate of the MFA algorithm has a certain fluctuation and tends to decline, while the classification performance of our RFA algorithm is relatively stable, which means that our algorithm is more robust than MFA. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Dataset** & **Classes** & **Samples** & **Dimensions** \\ \hline Ibn Sina & 174 & 20,688 & 200 \\ USPS & 10 & 9298 & 256 \\ Pendigits & 10 & 5620 & 16 \\ Optdigits & 10 & 10,992 & 64 \\ Letter & 26 & 20,000 & 16 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Statistics of the used datasets. Figure 1: One image of the Ibn Sina dataset. **USPS dataset:** The USPS is an U.S. post handwritten digits dataset that contains 7291 training data and 2007 test data from 10 classes and the dimensionality of the data features is 256. Some handwritten digits in the USPS dataset are shown in Figure 3.  
In this experiment, we set the number of the nearest neighbors - \(k\) of RFA to 24. The classification accuracies obtained by RFA and the compared algorithms are shown in Figure 4. Because the LDA subspace has a maximum of \(C-1\) dimension (\(C\) is the number of classes), LDA is only presented with a black star in the figure. We can see that when the dimension is 9, RFA obtains a comparative results with LDA and MFA. When the dimension increases, the results of RFA are always optimal. At the same time, the classification accuracy obtained by RDA is low. Figure 5 presents 3D visualization of the learned data representations by RFA, which shows the effect of RFA to obtain better classification boundaries between the classes. In Figure 6, we show the 2D projections of data learned by both RFA and MFA, to further show the effectiveness of RFA. It is easy to see that the samples processed by RFA are less likely to overlap at the boundary, indicating that compared to MFA, RFA preserves more properties that help distinguish the samples. In addition, the robustness of RFA is tested with respect to \(k_{i}\) and \(k_{p}\) (used to construct the intrinsic and penalty graphs). From Figure 4, it can be seen that RFA obtained the best result when the subspace dimension was 35, with parameter settings \(k_{i}=5\), \(k_{p}=20\) and \(k=24\). We fixed \(k\) and one of \(k_{i}\) and \(k_{p}\) to obtain the results when another parameter took different values. Figures 7 and 8 show that RFA is very robust. Figure 3: Sample images from the USPS dataset. Figure 2: Classification accuracy obtained by LDA, MFA and RFA on the Ibn Sina dataset. We also selected three document recognition-related datasets from the UCI machine learning repository to further test the effectiveness of RFA. They are Optdigits, Pendigits and Letter. Optdigits was preprocessed by NIST [57] programs to obtain 5620 instances in 8 \(\times\) 8 dimensions.