A Novel and Effective Method to Directly Solve Spectral Clustering

Spectral clustering has been attracting increasing attention due to its well-defined framework and excellent performance. However, most traditional spectral clustering methods consist of two separate steps: 1) Solving a relaxed optimization problem to learn the continuous clustering labels, and 2) Rounding the continuous clustering labels into discrete ones. The clustering results of the relax-and-discretize strategy inevitably result in information loss and unsatisfactory clustering performance. Moreover, the similarity matrix constructed from original data may not be optimal for clustering since data usually have noise and redundancy. To address these problems, we propose a novel and effective algorithm to directly optimize the original spectral clustering model, called Direct Spectral Clustering (DSC). We theoretically prove that the original spectral clustering model can be solved by simultaneously learning a weighted discrete indicator matrix and a structured similarity matrix whose connected components are equal to the number of clusters. Both of them can be used to directly obtain the final clustering results without any post-processing. Further, an effective iterative optimization algorithm is exploited to solve the proposed method. Extensive experiments performed on synthetic and real-world datasets demonstrate the superiority and effectiveness of the proposed method compared to the state-of-the-art algorithms.