Graph refining via iterative regularization framework

Graph-based methods have been widely applied in clustering problems. The mainstream pipeline for these methods is to build an affinity matrix first, and then use the spectral clustering methods to construct a graph. The existing studies about such a pipeline mainly focus on how to build a good affinity matrix, while the spectral method has only been considered as an end-up step to achieve the clustering tasks. However, the quality of the constructed graph has significant influences on the clustering results. Unlike most of the existing works, our studies in this paper focus on how to refine the original graph to construct a good graph by giving the number of clusters. We show that spectral clustering method has a good property of block structure preserving by giving the priori knowledge about number of clusters. Based on the property, we provide an iterative regularization framework to refine the original graph. The regularization framework is based on a well-designed reproducing kernel Hilbert spaces for vector-valued (RKHSvv) functions, which is in favor of doing kernel tricks on graph reconstruction. The elements in RKHSvv are multiple outputs affinity functions. We show that finding an optimal multiple outputs function is equivalent to construct a graph, and the associated affinity matrix of such a graph can be obtained in a form of multiplication between a kernel matrix and an unknown coefficient matrix.