# An Improved Evolutionary Multi-Objective Clustering Algorithm Based on Autoencoder
## 4 Results and Discussions
### The Influence of the Autoencoder  
The influence of the autoencoder is further discussed by comparing AE-IEMOKC with IEMO-KC. Note that \(p_{\text{m}}\) and \(\alpha\) for IEMO-KC are set to \(1/(d\cdot k_{\text{max}})\) and \(10/(m\cdot d)\), respectively, since the input datasets for IEMO-KC are the original datasets.  
Table 5 shows the ARI and ACC of IEMO-KC and AE-IEMOKC on the five datasets under their actual number of clusters. It can be observed that AE-IEMOKC is able to obtain higher ARI and ACC, especially on the Optdigits dataset. This directly demonstrates that the autoencoder can improve the clustering accuracy due to its ability to obtain the feature representation of the dataset suitable for clustering and its ability to reduce the dimensionality of the dataset. In fact, this improvement is not limited to the solution with the actual number of clusters. Taking the Iris dataset as an example, Figure 5 shows the ARI and ACC of the Pareto fronts obtained by IEMO-KC and AE-IEMOKC. It can be observed that the improvement in the accuracy of the autoencoder also works for some other solutions. However, it does not work for all the solutions, as the clustering loss in Equation (4) is obtained under the actual number of clusters. Overall, the autoencoder has a significant positive effect on the solutions, which have numbers of clusters close to the actual number of clusters.  
Figure 6 shows the clustering results obtained by IEMO-KC and AE-IEMOKC on the Iris dataset when \(k=2\), \(3\), \(4\). It can be clearly observed that AE-IEMOKC is able to achieve better clustering results under the actual number of clusters and its neighboring number of clusters, as the final \(H\) transformed from the original dataset by the autoencoder is more suitable for clustering. Similar results are observed for the other datasets.  
Figure 7 shows the running time averaged over 10 runs of EMO-KC, IEMO-KC1, IEMO-KC, and AE-IEMOKC on the five datasets. Each algorithm is implemented in Python 3.9 with a computer configuration of AMD R7-5800H CPU, 16 GB RAM, and RTX3050 4 GB GPU. It is observed that IEMO-KC1 consumes slightly more time than EMO-KC due to the additional computation of the scaling factor. However, the valid initialization has a greater influence than the scaling factor as IEMO-KC consumes significantly more time than IEMO-KC1. In fact, more time is not mainly consumed in the initialization process, but in the optimization, as the ensured validity of IEMO-KC makes the optimization more effective and thus more complex. It is observed that AE-IEMOKC consumes more time  
Figure 5: The ARI and ACC of the Pareto fronts obtained by IEMO-KC and AE-IEMOKC on the Iris dataset: (**a**) Comparison of ARI. (**b**) Comparison of ACC.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Metric** & **Algorithm** & **Iris** & **Wine** & **Seeds** & **BCW** & **Optdigits** \\ \hline \multirow{2}{*}{ARI} & IEMO-KC & 0.72 & 0.87 & 0.70 & 0.72 & 0.33 \\  & AE-IEMOKC & 0.87 & 0.90 & 0.77 & 0.79 & 0.67 \\ \hline \multirow{2}{*}{ACC} & IEMO-KC & 0.89 & 0.96 & 0.89 & 0.92 & 0.48 \\  & AE-IEMOKC & 0.95 & 0.97 & 0.92 & 0.95 & 0.79 \\ \hline \hline \end{tabular}
\end{table}
Table 5: The ARI and ACC of the clustering results obtained by IEMO-KC and AE-IEMOKC on the five datasets under their actual number of clusters.  
than IEMO-KC on the Iris, Wine, Seeds, and BCW datasets as the autoencoder part of AE-IEMOKC takes a lot of time to be pre-trained and fine-tuned, which greatly reduces the efficiency of AE-IEMOKC. However, the IEMO-KC part of AE-IEMOKC converges faster than IEMO-KC on the five datasets due to the autoencoder's ability to reduce the dimensionality, which is obvious on the BCW and Optdigits datasets. This suggests the autoencoder part of AE-IEMOKC is able to accelerate the convergence.