Robust Affine Subspace Clustering via Smoothed $\ell _{0}$-Norm

Abstract

In the past few years, sparse representation based method has been used in many fields with breathtaking speed due to its superior sparse recovery performance. Sparse subspace clustering (SSC), as one of its application hot-spots, has attracted considerable attention. Traditional sparse subspace clustering methods employ the $\ell _{1}$-norm to induce sparse representation of data points. Typically, the use of $\ell _{1}$-regularization instead of the $\ell _{0}$ one can make the objective function convex while it also causes large errors on large coefficients in some cases. In this work, we propose using the non-convex smoothed $\ell _{0}$-norm to replace the $\ell _{0}$ one for affine subspace clustering. This leads to a non-convex minimization problem. We then propose an effective method to solve the problem which minimizes the objective function by using the gradient method and proximal projection. In addition, the proposed algorithm is robust to noise and can provide a fast solution. Extensive experiments on real datasets demonstrate the effectiveness of our proposed method.