# Double-Constraint Fuzzy Clustering Algorithm
## 1 Introduction  
The c-means (CM) algorithm proposed by [PERSON] [1] is the most commonly used clustering algorithm over various research fields, but it cannot accurately partition data objects in which the membership to any specific cluster is uncertain [2]. As a general extension of CM, fuzzy clustering has been proposed to address this problem [3]. Most fuzzy clustering algorithms originate from [PERSON]'s fuzzy c-means (FCM) algorithm [4], which has been successfully applied in numerous applications including image segmentation [5], feature extraction [6], pattern recognition [7, 8], etc. However, the computed membership degrees in FCM are relative numbers. For a given data object, its membership degrees corresponding to all clusters in the fuzzy partition matrix must sum to 1 to avoid a trivial solution. The sum makes FCM noise-sensitive and unsuitable for applications in which membership degrees show a typicality or compatibility of data points with clusters under flexible constraints [9, 10]. Consequently, the clustering results of FCM are often inaccurate. To address these problems, various variants of FCM have been developed. [PERSON] et al. [11] proposed a possibilistic c-means method by giving up relative number constraints in FCM and thus presented the typicality or compatibility from data objects to clusters. [PERSON] et al. [12] assigned an importance degree to individual data points, thereby significantly mitigating the influence of relative numbers in FCM. More recently, numerous FCM-type algorithms have been proposed. [PERSON] [13] proposed a general c-means algorithm by extending the definition of means from statistical analysis. This algorithm generalized most variants of the fuzzy clustering method to a common model. [PERSON] et al. [14] proposed a feature-weighted c-means algorithm to address the difficulty that FCM faces in detecting clusters distributed across various subspaces. Despite progress, these algorithms mainly focus on finding clusters by the inherent distribution and distance among data objects [15, 16], but they cannot provide an assessmentutilizing the information on each data object and cluster. Especially, in various clustering applications, the number of data points in specific cluster is a mandatory constraint that the clustering results of any clustering algorithm must obey. If these constraints cannot be met, the clustering results may be unacceptable [17; 18]. To address this problem, [PERSON] et al. [19] proposed a constrained c-means (CCM) algorithm that assigns a fixed number of data points to each cluster, which means that each column in the partition matrix is constrained by an equation. Nevertheless, there are at least three unsolved problems in CCM. First, CCM must apply the solution of the transportation problem as a practical form, but this approach is not feasible for large-scale datasets due to high computational complexity. Secondly, CCM is a CM-type clustering algorithm, rather than a fuzzy clustering algorithm, and thus the issues that can be solved by FCM are not addressed by CCM. Finally, CCM cannot combine the importance degree of both clusters and data objects [20; 21]. More recently, efforts have been made to determine the number of clusters in the fuzzy clustering process. There are two approaches to determine the optimal number of clusters. One utilizes one or several clustering indices to determine the optimal number through a trial-and-error method across all possible numbers of clusters [22; 23]. The other attempts to solve the number of clusters in the iterative process of fuzzy clustering, such as the proposed Bayesian probabilistic model and inference algorithm for fuzzy clustering [24], which provides expanded capabilities compared to traditional FCM. However, these algorithms fail to address the problem of typicality or compatibility inherent in FCM. In this study, to address the typicality or compatibility of both data points and clusters, we incorporate dual constraints into each data point and cluster.