# Relational Fisher Analysis: Dimensionality Reduction in Relational Data with Global Convergence +
## 5 Experiments
### Performance of RFA  
To evaluate the performance of RFA, we selected several related dimensionality reduction methods for comparison with RFA. These methods were LDA, MFA, RDA, and PRPCA, respectively. Among them, LDA, MFA and RDA are supervised methods, while PRPCA is unsupervised. We note that, due to the flexibility in the design of the relational matrix, RFA could be either global or local. We describe this detail in the following part.  
We followed MFA to construct the intrinsic and penalty graphs [14]. The numbers of nearest neighbors for constructing the intrinsic graph (\(k_{i}\)) and the penalty graph (\(k_{p}\)) were set to 5 and 20, respectively, for all the datasets.  
We applied RFA to document understanding, face recognition and several other recognition tasks. In order to test the performance of RFA on document recognition tasks, we used the [PERSON] ancient Arabic document dataset [49], USPS handwritten digits dataset ([[http://www.cs.nyu.edu/~roweis/data.html](http://www.cs.nyu.edu/~roweis/data.html)]([http://www.cs.nyu.edu/~roweis/data.html](http://www.cs.nyu.edu/~roweis/data.html)), accessed on 19 October 2023), two handwritten digits datasets (Optdigits and Pendigits) and one English letter dataset (Letter) from the UCI machine learning repository [50]. For face recognition tasks, we used two face datasets [51; 52; 53; 54], the CMU PIE ([[http://www.face-rec.org/databases/](http://www.face-rec.org/databases/)]([http://www.face-rec.org/databases/](http://www.face-rec.org/databases/)), accessed on 19 October 2023) and the YaleB ([[http://www.cad.zju.edu.cn/home/dengcai/Data/FaceData.html](http://www.cad.zju.edu.cn/home/dengcai/Data/FaceData.html)]([http://www.cad.zju.edu.cn/home/dengcai/Data/FaceData.html](http://www.cad.zju.edu.cn/home/dengcai/Data/FaceData.html)), accessed on 19 October 2023) datasets. At the same time, we used some UCI datasets, including Shuttle, Thyroid, Vowel and Waveform21, to evaluate RFA.  
In our experiments, we used the classical graph Laplacian matrix, \(\mathbf{L}=\mathbf{D}-\mathbf{M}\), to define the relational matrix, whose weight matrix \(\mathbf{M}\) is shown as below:  
\[\mathbf{M}_{ij}=\begin{cases}1,&\text{ if }\mathbf{x}_{i}\in\mathcal{N}_{k}( \mathbf{x}_{j})\text{ or }\mathbf{x}_{j}\in\mathcal{N}_{k}(\mathbf{x}_{i}),\\ 0,&\text{ otherwise,}\end{cases} \tag{24}\]  
where \(\mathcal{N}_{k}(\mathbf{x}_{i})\) is the set consisting of the \(k\) nearest neighborForof \(\mathbf{x}_{i}\). For each dataset, the value of \(k\) was selected based on 5-fold cross-validation. Moreover, \(\mathbf{D}\) is the diagonal degree matrix with \(\mathbf{D}_{ii}=\sum_{j}\mathbf{M}_{ij}\).  
We note that the Laplacian weight matrix formed by the above formula contains the relationship information between the sample and a certain number of its neighbors, which allows the integration of the local relationships between data into the supervised representation learning algorithm. At the same time, as a general model, the relationship matrix \(\mathbf{R}\) in RFA can be of various forms. For example, \(\mathbf{R}\) can be the centralization matrix \(\mathbf{H}=\mathbf{I}_{N}-\frac{1}{N}\mathbf{1}\mathbf{1}^{T}\), where \(N\) denotes the data size and \(\mathbf{1}\) is a column vector of length \(N\) with all ones.  
For the PRPCA algorithm, we performed this experiment using the codes provided by the authors. For the RDA algorithm, we randomly selected the prototypes [24]. For algorithms other than RDA, we used the 1-nearest neighbor classifier to evaluate the classification performance of them.