# Relational Fisher Analysis: Dimensionality Reduction in Relational Data with Global Convergence +
## 5 Experiments
### Performance of RFA
#### 5.1.4 Comparison on Document Classification and Webpage Classification Problems  
As a general dimensionality reduction framework, the relational matrix can be constructed with different strategies. In the previous sections, we considered the relationship between samples based on their class labels or similarity. However, in some complicated problems, relationships may presented in other forms. For example, as indicated in [25], if there is a reference relationship between two papers, they are likely to have the same topic. However, due to the sparse nature of the bag-of-words representation, the similarity between these two papers may be very low. Thus, to further testify RFA, we designed a relational matrix based on the citation relevance between data samples to model RFA, and tested its effectiveness on document classification and webpage classification problems.  
For this experiment, we used two datasets, citeseer and WebKB ([[https://liqs-data.soe.ucsc.edu/public/lbc/](https://liqs-data.soe.ucsc.edu/public/lbc/)]([https://liqs-data.soe.ucsc.edu/public/lbc/](https://liqs-data.soe.ucsc.edu/public/lbc/)), accessed on 19 October 2023). We note that the WebKB dataset contains four subsets: Cornell, Texas, Washington and Wisconsin, and we show the experimental results of these four subsets separately. Each dataset contains bag-of-words representation of documents or webpages and citation links between the instances. Citeseer contains 3312 scientific documents from 6 different classes, and there are 4732 citation relation between the documents. WebKB consists of 877 webpages from 5 different classes, and there are 1608 page links within this dataset. We adopted the same strategy as in PRPCA to construct the relational matrix:  
(1) Constructing the adjacent graph \(\mathbf{A}\) according to the relevance between data samples. If there was a citation or link between sample \(i\) and \(j\), then \(\mathbf{A}_{ij}=1\); else, \(\mathbf{A}_{ij}=0\).  
(2) Letting \(\tilde{\mathbf{D}}_{ii}=\sum_{j}\mathbf{A}_{ij}=(\mathbf{A}\mathbf{A})_{ij}\), then \(\mathbf{B}=\mathbf{A}\mathbf{A}-\tilde{\mathbf{D}}\),  
\[\mathbf{B}_{ij}=\begin{cases}(\mathbf{A}\mathbf{A})_{ij}=\sum_{k=1}^{N} \mathbf{A}_{ij}\mathbf{A}_{kj},&\text{if }i\
eq j,\\ 0,&\text{otherwise}.\end{cases} \tag{25}\]  
(3) Defining \(\mathbf{G}=2\mathbf{A}+\mathbf{B}\) as the relational matrix in RFA.  
We took PRPCA as the baseline method in this part. Experimental results are illustrated in Figure 11; we can see that RFA achieves comparable results with PRPCA in all these five datasets and is even better than PRPCA on some of the datasets.  
\begin{table}
\begin{tabular}{c c c c c c} \hline
**Dimension** & **LDA** & **MFA** & **PRPCA** & **RDA** & **RFA** \\ \hline Shuttle & 0.9975 \(\pm\) 0.0009 & 0.9979 \(\pm\) 0.0012 & 0.9959 \(\pm\) 0.0037 & 0.9162 \(\pm\) 0.0132 & **0.9981 \(\pm\) 0.0003** \\ Thyroid & 0.9395 \(\pm\) 0.0265 & 0.9488 \(\pm\) 0.0303 & 0.9488 \(\pm\) 0.0195 & 0.9209 \(\pm\) 0.0265 & **0.9581 \(\pm\) 0.0255** \\ Vowel & 0.9859 \(\pm\) 0.0109 & 0.9869 \(\pm\) 0.0085 & 0.9859 \(\pm\) 0.0090 & 0.3505 \(\pm\) 0.0349 & **0.9879 \(\pm\) 0.0077** \\ Waveform21 & 0.8184 \(\pm\) 0.0119 & 0.8258 \(\pm\) 0.0197 & 0.8150 \(\pm\) 0.0118 & 0.5972 \(\pm\) 0.0279 & **0.8278 \(\pm\) 0.0175** \\ \hline \end{tabular}
\end{table}
Table 9: Classification results obtained on several UCI datasets.