# An Improved Evolutionary Multi-Objective Clustering Algorithm Based on Autoencoder
## 4 Results and Discussions  
### The Accuracy of AE-IEMOKC  
The accuracy of AE-IEMOKC is demonstrated by comparing with EMO-KC [7], GKA [21], and MOKGA [22]. Furthermore, the population size \(N\), and the _maxgen_, crossover, and mutation operators for EMO-KC, GKA, and MOKGA are kept consistent with those in Section 3.2. Values of \(p_{\text{m}}\) for EMO-KC, GKA, and MOKGA are set to \(1/(d\cdot k_{\text{max}})\) as the input datasets for them are the original datasets instead of the final \(H\).  
To briefly demonstrate the clustering accuracy of AE-IEMOKC, a solution corresponding to the actual number of clusters of each dataset is selected from the set of clustering solutions obtained as an example for explanation. Table 3 shows the obtained ARI and ACC. It can be clearly observed that EMO-KC has the smallest ARI and ACC on all the datasets due to its invalidity. It is evident that AE-IEMOKC has the highest ARI and ACC on all the datasets, which means that the best accuracy solutions can be obtained by our proposed algorithm. Specifically, the largest improvement in AE-IEMOKC's clustering accuracy over GKA and MOKGA is on the Optdigits dataset. It is difficult for both GKA and MOKGA to converge on the dataset Optdigits due to the high dimensionality, which results in numerous decision variables and a large search space. However, AE-IEMOKC converges easily due to the autoencoder's ability to reduce the dimensionality.  
\begin{table}
\begin{tabular}{c c c} \hline
**Dataset** & **Learning Rate** & \(\lambda\) \\ \hline Iris & \(1\times 10^{-3}\) & \(1\times 10^{-3}\) \\ Wine & \(1\times 10^{-3}\) & \(5\times 10^{-2}\) \\ Seeds & \(5\times 10^{-5}\) & \(1\times 10^{1}\) \\ BCW & \(7\times 10^{-4}\) & \(5\times 10^{-2}\) \\ Optdigits & \(1\times 10^{-3}\) & \(1\times 10^{-2}\) \\ \hline \end{tabular}
\end{table}
Table 2: Settings of the learning rate and the parameter \(\lambda\) for different datasets.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline
**Metric** & **Algorithm** & **Iris** & **Wine** & **Seeds** & **BCW** & **Optdigits** \\ \hline \multirow{4}{*}{ARI} & EMO-KC & 0.51 & 0.29 & 0.47 & 0.04 & 0.20 \\  & GKA & 0.72 & 0.85 & 0.70 & 0.73 & 0.37 \\  & MOKGA & 0.72 & 0.85 & 0.70 & 0.73 & 0.37 \\  & AE-IEMOKC & 0.87 & 0.90 & 0.77 & 0.79 & 0.67 \\ \hline \multirow{4}{*}{ACC} & EMO-KC & 0.67 & 0.64 & 0.74 & 0.66 & 0.38 \\  & GKA & 0.89 & 0.95 & 0.89 & 0.93 & 0.57 \\ \cline{1-1}  & MOKGA & 0.89 & 0.95 & 0.89 & 0.93 & 0.57 \\ \cline{1-1}  & AE-IEMOKC & 0.95 & 0.97 & 0.92 & 0.95 & 0.79 \\ \hline \end{tabular}
\end{table}
Table 3: The ARI and ACC of the clustering results obtained by EMO-KC, GKA, MOKAG, and AE-IEMOKC on the five datasets under their actual number of clusters.  
Figure 3 shows the clustering results obtained by GKA, MOKGA, and AE-IEMOKC on the Iris, Wine, and Seeds datasets when \(k=3\), where the clustering results obtained by EMO-KC are not shown due to their invalidity. The mark (+) denotes the cluster centroid. It can be observed that the distribution of the data points in AE-IEMOKC is significantly different from that in GKA and MOKGA, as the data points in AE-IEMOKC are actually the final \(H\) transformed from the original datasets by the autoencoder. Since the final \(H\) is more suitable for clustering, AE-IEMOKC is able to achieve better clustering results, as it can result in tighter data points within the same cluster and clearer distinctions between data points within different clusters. Similar results are also observed for the BCW and Optdigits datasets.