# One-Shot Distributed Generalized Eigenvalue Problem (DGEP): Concept, Algorithm and Experiments +
## 5 Numerical Experiments
### Distributed CCA on Real Datasets  
In this case, we apply Algorithm 3 to a multi-classification problem where the feature is denoted as a view \(X\) and the class as another view \(Y\). Two real datasets from the gene expression database [35] are considered as data-sensitive and are dispersed evenly for convenience in local servers. Each dataset is locally divided into training data and testing data in terms of the same data dimension \(d\). The details are explained below, and the statistics can be found in Table 3:  
* Lymphoma: 42 samples of diffuse large B-cell lymphoma, 9 observations of follicular lymphoma, and 11 cases of chronic;
* SRBCT: the filtered dataset of 2308 gene expression profiles for 4 types of small round blue cell tumors of childhood. The classification accuracy is defined as  
\[acc=\frac{\sum_{i=1}^{num}\delta(ol_{i},pl_{i})}{num},\]  
where \(\delta(ol_{i},pl_{i})\) is the indicator function. When the obtained label \(ol_{i}\) is equal to the provided label \(pl_{i}\), it equals 1 and 0 otherwise. Considering the limited number of Gene data, the maximum \(N\) is set to be 8. The experiments on two gene datasets are repeated 20 times each, and the detailed mean accuracy is reported in Table 4. It reveals that the accuracy decreases as the number of local servers \(N\) goes up. The divergence of covariance in the dis \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Method** & & **Accuracy** & **Mean Accuracy** \\ \hline \multirow{4}{*}{Centered by SVD} & \(acc_{C_{1}}tr\) & 1.0000 + 0.0000 & \multirow{4}{*}{1.0000 for train} \\ \cline{2-4}  & \(acc_{C_{2}}tr\) & 1.0000 + 0.0000 & & \\ \cline{2-4}  & \(acc_{C_{1}}ts\) & 0.9835 + 0.0250 & \multirow{4}{*}{0.9706 for test} \\ \cline{2-4}  & \(acc_{C_{2}}ts\) & 0.9576 + 0.0654 & & \\ \hline \multirow{4}{*}{DGEP-SVD} & \(acc_{C_{1}}tr\) & 0.9600+0.0711 & \multirow{4}{*}{0.9271 for train} \\ \cline{2-4}  & \(acc_{C_{2}}tr\) & 0.8943 + 0.0895 & & \\ \cline{2-4}  & \(acc_{C_{1}}ts\) & 0.9407 + 0.0990 & & \\ \cline{2-4}  & \(acc_{C_{2}}ts\) & 0.8593 + 0.1052 & & \\ \hline \multirow{4}{*}{DGEP-PM} & \(acc_{C_{1}}tr\) & 0.9578+0.0477 & \multirow{4}{*}{0.9326 for train} \\ \cline{2-4}  & \(acc_{C_{2}}tr\) & 0.9075 + 0.0625 & & \\ \cline{1-1} \cline{2-4}  & \(acc_{C_{1}}ts\) & 0.9505 + 0.0526 & & \\ \cline{1-1} \cline{2-4}  & \(acc_{C_{2}}ts\) & 0.8741 + 0.0745 & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Classification accuracy of GSE2187 data in FDA: training accuracy of \(C_{i}\) (\(acc_{C_{i}}tr\)), testing accuracy of \(C_{i}\) (\(acc_{C_{i}}ts\)). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Type** & **Data** & \(K\) & \(d\) & \(num\) & \(tr_{num}\) & \(ts_{num}\) \\ \hline \multirow{2}{*}{Gene Data} & Lymphoma & 3 & 4026 & 62 & 48 & 14 \\  & SRBCT & 4 & 2308 & 63 & 48 & 15 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Genedata structures: data dimension (\(d\)), number of data (\(num\)), number of training data (\(tr_{num}\)), number of testing data (\(tr_{num}\)), number of testing data (\(ts_{num}\)), number of classes (\(K\)). tributed way influences the classification accuracy of Gene data, especially in SRBCT. However, it is acceptable especially in multi-classification. When \(N=1\), without the communication error, the results reveal that DGEP-PM, the distributed CCA in GEP form without an eigengap, leads to a worse accuracy which is no higher than 72% for training and 68% for testing in Lymphoma and no higher than 41% for training and 36% for testing in SRBCT. The performance of Algorithm 3 in Table 4 is obviously higher. To illustrate the difference in detail, we further investigate the convergence and the error of the two algorithms and show the result in Figure 5. For Lymphoma data as an instance, we define _convergence_ as the successive difference of the objective function value, and \(Err\) as the error between the objective function value of the proposed method and ground truth. The maximum iteration is set to be 10.