# An Improved Evolutionary Multi-Objective Clustering Algorithm Based on Autoencoder
## 4 Results and Discussions
### The Validity of AE-IEMOKC  
The validity of AE-IEMOKC is ensured by its IEMO-KC part. To demonstrate the validity of AE-IEMOKC, this subsection makes a comparison among EMO-KC, IEMO-KC1 (EMO-KC based on the scaling factor), IEMO-KC2 (EMO-KC based on the valid initialization), and IEMO-KC, without considering the autoencoder. Note that the values of \(p_{\text{m}}\) for the four algorithms are set to \(1/(d\cdot k_{\text{max}})\), and \(\alpha\) for both IEMO-KC1 and IEMO-KC is set to \(10/(m\cdot d)\), since the input datasets for them are the original datasets.  
Figure 3: Clustering results obtained by GKA, MOKGA, and AE-IEMOKC on the Iris, Wine, and Seeds datasets when \(k=3\): (**a**) The clustering results obtained by GKA (**b**) The clustering results obtained by MOKGA. (**c**) The clustering results obtained by AE-IEMOKC.  
Table 4 shows the SSD and \(f_{1}\) obtained by the four algorithms on the five datasets under their actual number of clusters. It shows that the SSD of EMO-KC is the largest, and the \(f_{1}\) of EMO-KC is approximately equal to \(-2.00\), which shows the poor validity of EMO-KC. However, the SSD and \(f_{1}\) of IEMO-KC1 are smaller than those of EMO-KC, which suggests that the scaling factor can help effectively optimize the objective of SSD\({}_{\text{exp}}\). It can also be observed that the SSD of IEMO-KC2 is smaller than those of EMO-KC and IEMO-KC1. This is because in the valid initialization of IEMO-KC2, data points rather than points in the search space are selected as the cluster centroids, which allows the cluster centroids to be closer to the other data points, resulting in a smaller SSD. However, the \(f_{1}\) of IEMO-KC2 is approximately equal to \(-2.00\) due to the lack of the scaling factor, indicating that the objective of SSD\({}_{\text{exp}}\) has not been effectively optimized. However, it is evident that IEMO-KC is able to obtain the smallest SSD and \(f_{1}\) simultaneously, which shows that the combination of the scaling factor and valid initialization in IEMO-KC contributes to the validity.  
The validity is further demonstrated using the Wine dataset as an example. Figure 4 shows the Pareto fronts of SSD and \(f_{1}\) obtained by the four algorithms. It can be observed that IEMO-KC2 and IEMO-KC are able to obtain clustering solutions with all different \(k\). This indicates that valid initialization is able to avoid invalid solutions. It can also be clearly observed that all the solutions obtained by EMO-KC, IEMO-KC1, and IEMO-KC2 are Pareto-dominated by those obtained by IEMO-KC, which further shows that the validity is ensured by the combination of the scaling factor and valid initialization. Similar results are observed for other datasets.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Metric** & **Algorithm** & **Iris** & **Wine** & **Seeds** & **BCW** & **Optdigits** \\ \hline \multirow{4}{*}{SSD} & EMO-KC & 34.89 & 259.38 & 145.79 & 2507.83 & 23,678.61 \\  & IEMO-KC1 & 31.10 & 182.02 & 128.60 & 1902.68 & 23,646.44 \\  & IEMO-KC2 & 10.09 & 88.47 & 117.32 & 423.08 & 9137.39 \\  & IEMO-KC & 6.98 & 49.00 & 22.03 & 216.25 & 7488.99 \\ \hline \multirow{4}{*}{\(f_{1}\)} & EMO-KC & \(-2.00\) & \(-2.00\) & \(-2.00\) & \(-2.00\) & \(-2.00\) \\  & IEMO-KC1 & \(-2.60\) & \(-2.46\) & \(-2.42\) & \(-2.33\) & \(-2.13\) \\  & IEMO-KC2 & \(-2.00\) & \(-2.00\) & \(-2.00\) & \(-2.00\) & \(-2.00\) \\  & IEMO-KC & \(-2.89\) & \(-2.81\) & \(-2.86\) & \(-2.88\) & \(-2.52\) \\ \hline \hline \end{tabular}
\end{table}
Table 4: The SSD and \(f_{1}\) of the clustering results obtained by EMO-KC, IEMO-KC1, IEMO-KC2, and IEMO-KC on the five datasets under their actual number of clusters.  
Figure 4: Pareto fronts obtained by the four algorithms on the Wine dataset, where solutions containing the invalid clusters are removed due to their invalidity: (**a**) Comparison of SSD. (**b**) Comparison of \(f_{1}\).