A Survey on Soft Subspace Clustering <s> (2) FWFKM <s> In fuzzy clustering algorithms each object has a fuzzy membership associated with each cluster indicating the degree of association of the object to the cluster. Here we present a fuzzy subspace clustering algorithm, FSC, in which each dimension has a weight associated with each cluster indicating the degree of importance of the dimension to the cluster. Using fuzzy techniques for subspace clustering, our algorithm avoids the difficulty of choosing appropriate cluster dimensions for each cluster during the iterations. Our analysis and simulations strongly show that FSC is very efficient and the clustering results produced by FSC are very high in accuracy. <s> BIB001 </s> A Survey on Soft Subspace Clustering <s> (2) FWFKM <s> We establish the convergence of the fuzzy subspace clustering (FSC) algorithm by applying Zangwill's convergence theorem. We show that the iteration sequence produced by the FSC algorithm terminates at a point in the solution set S or there is a subsequence converging to a point in S. In addition, we present experimental results that illustrate the convergence properties of the FSC algorithm in various scenarios. <s> BIB002 </s> A Survey on Soft Subspace Clustering <s> (2) FWFKM <s> K-means is one of the most popular and widespread partitioning clustering algorithms due to its superior scalability and efficiency. Typically, the K-means algorithm treats all features fairly and sets weights of all features equally when evaluating dissimilarity. However, a meaningful clustering phenomenon often occurs in a subspace defined by a specific subset of all features. To address this issue, this paper proposes a novel feature weight self-adjustment (FWSA) mechanism embedded into K-means in order to improve the clustering quality of K-means. In the FWSA mechanism, finding feature weights is modeled as an optimization problem to simultaneously minimize the separations within clusters and maximize the separations between clusters. With this objective, the adjustment margin of a feature weight can be derived based on the importance of the feature to the clustering quality. At each iteration in K-means, all feature weights are adaptively updated by adding their respective adjustment margins. A number of synthetic and real data are experimented on to show the benefits of the proposed FWAS mechanism. In addition, when compared to a recent similar feature weighting work, the proposed mechanism illustrates several advantages in both the theoretical and experimental results. <s> BIB003
The Feature Weighted Fuzzy K-means (FWFKM) algorithm performs clustering through an iterative procedure based on the fuzzy k-means algorithm and the supervised ReliefF algorithm  . With D as the number of features, the FWFKM algorithm begins by setting the weights as 1/ D and implementing the fuzzy k-means algorithm with the weighted distances to obtain the initial clustering result and label the data. With the labeled data, the supervised ReliefF algorithm is then used to assign the new weights for every feature. This procedure is conducted iteratively with the weights updated repeatedly until the final clustering result is achieved.  With the current partition in the iterative k-means clustering process, the W-k-means algorithm calculates a new weight for each variable, i.e., feature, based on the variance of the within-cluster distances. The new weights are used to decide the cluster memberships of objects in the next iteration. The optimal weights are found when the algorithm converges, which can then be used to identify important features for clustering. The features that may indeed be noise to the clustering process can be removed in future analysis. The W-k-means algorithm has received increasing attention, based on which modified algorithms are proposed, such as the fuzzy subspace clustering algorithm BIB001 BIB002 to be reviewed in Subsection 4.1.1. BIB003 . The algorithm adopts the objective function below, Furthermore, the following sub-optimization function is used to adjust the weights where || || i C denotes the number of the i th cluster obtained in the current iteration and is the global center of all data objects in the data set. The final clustering results and weights are then obtained iteratively. 