On Tolerant Fuzzy c-Means Clustering with L1-Regularization

We have proposed tolerant fuzzy c-means clustering (TFCM) from the viewpoint of handling data more flexibly. This paper presents a new type of tolerant fuzzy c-means clustering with L1-regularization. L1-regularization is wellknown as the most successful techniques to induce sparseness. The proposed algorithm is different from the viewpoint of the sparseness for tolerance vector. In the original concept of tolerance, a tolerance vector attributes to each data. This paper develops the concept to handle data flexibly, that is, a tolerance vector attributes not only to each data but also each cluster. First, the new concept of tolerance is introduced into optimization problems. These optimization problems are based on conventional fuzzy c-means clustering (FCM). Second, the optimization problems with tolerance are solved by using Karush-Kuhn-Tucker conditions and an optimization method for L1-regularization. Third, new clustering algorithms are constructed based on the explicit optimal solutions. Finally, the effectiveness of the proposed algorithm is verified through some numerical examples. Keywords— fuzzy c-means clustering, L1-regularization, optimization, tolerance, uncertainty