New _2,1 -norm relaxation of multi-way graph cut for clustering

The clustering methods have absorbed even-increasing attention in machine learning and computer vision communities in recent years. Exploring manifold information in multi-way graph cut clustering, such as ratio cut clustering, has shown its promising performance. However, traditional multi-way ratio cut clustering method is NP-hard and thus the spectral solution may deviate from the optimal one. In this paper, we propose a new relaxed multi-way graph cut clustering method, where 2,1-norm distance instead of squared distance is utilized to preserve the solution having much more clearer cluster structures. Furthermore, the resulting solution is constrained with normalization to obtain more sparse representation, which can encourage the solution to contain more discrete values with many zeros. For the objective function, it is very difficult to optimize due to minimizing the ratio of two non-smooth items. To address this problem, we transform the objective function into a quadratic problem on the Stiefel manifold (QPSM), and introduce a novel yet efficient iterative algorithm to solve it. Experimental results on several benchmark datasets show that our method significantly outperforms several state-of-the-art clustering approaches.