# An Improved Evolutionary Multi-Objective Clustering Algorithm Based on Autoencoder
###### Abstract
If SSD is large, the first term of \(\text{SSD}_{\text{exp}}\) (see Equation (5)), \(1-\exp^{-1\cdot\text{SSD}}\), would be approximately equal to 1, and the second term of \(\text{SSD}_{\text{exp}}\), \(-k\), would dominate \(\text{SSD}_{\text{exp}}\). Thus, different clustering solutions with the same \(k\) in the population will obtain almost the same \(\text{SSD}_{\text{exp}}\), making it difficult to optimize the objective of \(\text{SSD}_{\text{exp}}\). As a result, EMO-KC's validity is limited. In addition, the treatment of invalid solutions is not considered in the initialization process of EMO-KC. Points in the search space are randomly selected as the cluster centroids and encoded into the chromosomes representing the clustering solutions in EMO-KC. Thus, there may be some invalid clusters without any data points in the clustering solution for a certain \(k\), making the solution unrealistic as its number of valid clusters is less than \(k\). As a result, it is not ensured to obtain the clustering solutions with all different \(k\), which also limits the validity of EMO-KC. Previous studies [8; 9; 10; 11; 12; 13] have focused on the above issues of EMO-KC. A reduced-length chromosome encoding method was used to reduce the number of decision variables in [9], which was also commonly used in earlier studies to improve the clustering accuracy, especially on high-dimensional datasets [10; 11; 12]. The number of features representing the dimensionality of the datasets was taken as an optimizing objective in [13] to reduce the number of decision variables. However, the dimension reduction of the input datasets has rarely been considered. For this issue, autoencoder as a data dimension reduction method based on deep learning has gained popularity in clustering and helps to obtain good clustering accuracy especially on high-dimensional datasets [14; 15; 16; 17; 18; 19; 20]. It can maintain the nonlinear feature of the datasets while reducing the dimensionality. [PERSON] et al. [14] used the autoencoder to reduce the dimensionality of the datasets by mapping the datasets to the low-dimensional embedding layer of the autoencoder as the feature representation. Then, the feature representation was clustered by k-means, which could significantly improve the accuracy of clustering. The following studies have focused on the expressions of the autoencoder's loss functions and the clustering algorithms used [15; 16; 17; 18; 19; 20]. Thus, EMO-KC as a clustering algorithm is expected to obtain better clustering accuracy when combined with the autoencoder. However, the combination of EMO-KC and the autoencoder has not been proposed in related studies. Furthermore, [PERSON] et al. [9] analyzed the invalidity of EMO-KC, i.e., the objective of \(\text{SSD}_{\text{exp}}\) could not be effectively optimized. However, no measures were taken to address this issue in their study and other relevant studies. In addition, to reduce the influence of the invalid clusters, a constrained decomposition based on grids (CDG) was introduced into CCDG-K [8] to divide the clustering task into multiple subtasks, each of which focused on optimizing the single objective of \(\text{SSD}_{\text{exp}}\). This ensured that clustering solutions could be obtained for all different \(k\). However, the treatment of the invalid solutions was still not considered in CCDG-K. Actually, an effective approach to avoid the invalid solutions is to select data points as the cluster centroids and encode them into the chromosome during the initialization process (called valid initialization for short) as in GKA [21] and MOKGA [22]. However, valid initialization has not been considered in EMO-KC yet. In this paper, an improved evolutionary multi-objective clustering algorithm based on autoencoder (AE-IEMOKC) is proposed to improve the accuracy and ensure the validity of EMO-KC.