Hyperparameter-Free Localized Simple Multiple Kernel K-Means With Global Optimum

The newly proposed localized simple multiple kernel k-means (SimpleMKKM) provides an elegant clustering framework which sufficiently considers the potential variation among samples. Although achieving superior clustering performance in some applications, we observe that it is required to pre-specify an extra hyperparameter, which determines the size of the localization. This greatly limits its availability in practical applications since there is a little guideline to set a suitable hyperparameter in clustering tasks. To overcome this issue, we firstly parameterize a neighborhood mask matrix as a quadratic combination of a set of pre-computed base neighborhood mask matrices, which corresponds to a group of hyperparameters. We then propose to jointly learn the optimal coefficient of these neighborhood mask matrices together with the clustering tasks. By this way, we obtain the proposed hyperparameter-free localized SimpleMKKM, which corresponds to a more intractable minimization-minimization-maximization optimization problem. We rewrite the resultant optimization as a minimization of an optimal value function, prove its differentiability, and develop a gradient based algorithm to solve it. Furthermore, we theoretically prove that the obtained optimum is the global one. Comprehensive experimental study on several benchmark datasets verifies its effectiveness, comparing with several state-of-the-art counterparts in the recent literature. The source code for hyperparameter-free localized SimpleMKKM is available at https://github.com/xinwangliu/SimpleMKKMcodes/.