Schatten-q regularizer constrained low rank subspace clustering model

Abstract

In the global low rank spectral subspace clustering model, the rank minimization problem is relaxed as Nuclear Norm Minimization (NNM) problem. This relaxation is widely used due to its convexity. However, in recent years, the non-convex regularization has become widely used in signal recovery, matrix completion, and pattern analysis. A powerful tool for the non-convex regularization in the subspace clustering model, the Schatten-q regularizer is relatively unexplored. In this paper, we introduce the non-convex Schatten-q regularizer for the subspace clustering problem in order to solve the rank minimization problem. In this context, we present the GMST algorithm, a new generalized matrix soft thresholding algorithm, to solve the Schatten-q regularizer minimization problem. The proposed method always obtains a solution with a lower rank than the other methods. This shows that the GMST algorithm has the ability to depict the structure of the redundant data to a much greater extent than the existing methods. A large number of experiments demonstrate that the proposed method is competitive to the state-of-the-art methods, but has a lower computational cost and is especially more robust to outliers. Furthermore, our newly proposed solver to Schatten-q (0<q<1) regularizer is more accurate. Many current solvers to Schatten-q regularizer have reported that when q=1, their methods will become the widely used singular value thresholding algorithm. Beyond that, when q=0.5, our newly proposed solver also coincides with S1/2 regularizer based half thresholding algorithm. A rigorous mathematical proof is given.