## 5 Discussion and Conclusions  
In this study, we have proposed a tree-based data reduction approach for solving large-scale SVM problems. In order to reduce time consumption in training SVM models, we apply a novel support vector selection method combining tree decomposition and the proposed relative support distance. We introduce the relative distance measure along with a virtual hyperplane between two distinct adjacent regions to effectively exclude non-SV data points. The virtual hyperplane, easily obtainable, takes advantage of the decomposed tree structures and is shown to be effective in selecting support vector candidates. In computing the relative support distance, we also use the distance between each data point to the centroid in each region and combine the two in consideration of the nonlinear characteristics of support vectors. In experiments, we have demonstrated that the proposed method outperforms some existing methods for selecting support vector candidates in terms of computation time and classification performance. In the future, we would like to investigate other large-scale  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Dataset**} & \multicolumn{2}{c}{**SVM**} & \multicolumn{2}{c}{**CVS**} & \multicolumn{2}{c}{**FLD**} & \multicolumn{2}{c}{**TRSD**} \\ \cline{2-9}  & **SC** & **TS** & **SC** & **TS** & **SC** & **TS** & **SC** & **TS** \\ \hline Iris-Setosa & 0 & 0.01 & 0 & 0.01 & 0 & 0.01 & 0 & 0.01 \\ Iris-Versicolor & 0 & 0.01 & 0 & 0.01 & 0 & 0.01 & 0 & 0.01 \\ Iris-Virginia & 0 & 0.01 & 0 & 0.01 & 0 & 0.01 & 0 & 0.01 \\ Breast Cancer & 0 & 0.01 & 0 & 0.01 & 0.07 & 0.01 & 0.04 & 0.01 \\ Four-class & 0 & 0.01 & 0 & 0.01 & 0.06 & 0.01 & 0.04 & 0.01 \\ Checkerboard & 0 & 0.03 & 0 & 0.01 & - & - & 0.09 & 0.01 \\ German Credit & 0 & 0.09 & 0.01 & 0.02 & 0.33 & 0.02 & 0.27 & 0.02 \\ Waveform-0 & 0 & 1.03 & 0.18 & 0.17 & 0.58 & 0.13 & 0.3 & 0.11 \\ Waveform-1 & 0 & 0.68 & 0.17 & 0.17 & 0.65 & 0.09 & 0.3 & 0.08 \\ Waveform-2 & 0 & 0.94 & 0.17 & 0.16 & 0.73 & 0.11 & 0.34 & 0.07 \\ Banana & 0 & 0.59 & 0 & 0.17 & 0.08 & 0.04 & 0.05 & 0.05 \\ Mushroom & 0 & 1.47 & 1.9 & 0.27 & 0.99 & 0.72 & 0.71 & 0.26 \\ Phishing & 0 & 4.59 & 2.08 & 0.92 & 1.3 & 0.69 & 0.72 & 0.57 \\ w8a & 0 & 114.85 & 17.48 & 32.15 & 228.83 & 8.18 & 46.35 & 8.74 \\ a9a & 0 & 373.24 & 44.5 & 82.88 & 16.52 & 60.66 & 8.84 & 34.18 \\ IJCNN-1 & 0 & 224.29 & 6.06 & 31.73 & 10.08 & 24.67 & 5.77 & 19.66 \\ Skin Segmentation & 0 & 20.23 & 0.3 & 3.54 & 3.22 & 3.79 & 2.3 & 1.48 \\ Cod-RNA & 0 & 4425.55 & 0.7 & 898.4 & 15.46 & 582.77 & 11.24 & 245.47 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Time comparison in detail. SC and TS mean computing time for selecting candidate vectors and training a final SVM model respectively.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(\beta\) & **TRSD \textgreater{} CVS** & **TRSD \(\approx\) CVS** & **TRSD \textless{} CVS** & **TRSD \textgreater{} FLD** & **TRSD \(\approx\) FLD** & **TRSD \textless{} FLD** \\ \hline
0.1 & 10 & 6 & 2 & 9 & 9 & 0 \\
0.2 & 11 & 4 & 3 & 9 & 8 & 1 \\
0.3 & 11 & 4 & 3 & 10 & 8 & 0 \\ Total & 32 & 14 & 8 & 28 & 25 & 1 \\ \hline \hline \end{tabular}
\end{table}
Table 5: [PERSON]’s test results in different \(\beta\) for the significant level \(\alpha=0.05\).  
SVM problems such as multi-class classification and support vector regression. We also envision an extension of the proposed method to under-sampling techniques.  
Investigation, M.R.; Methodology, M.R.; Software, M.R.; Writing--original draft, M.R.; Writing--review & editing, K.L. All authors have read and agreed to the published version of the manuscript. This research was supported by the Ministry of Education of the Republic of Korea and the National Research Foundation of Korea (NRF-2020R1F1A1076278). The authors declare no conflict of interest.