Robust Subspace Clustering via Smoothed Rank Approximation

Matrix rank minimizing subject to affine constraints arises in many application areas, ranging from signal processing to machine learning. Nuclear norm is a convex relaxation for this problem which can recover the rank exactly under some restricted and theoretically interesting conditions. However, for many real-world applications, nuclear norm approximation to the rank function can only produce a result far from the optimum. To seek a solution of higher accuracy than the nuclear norm, in this letter, we propose a rank approximation based on Logarithm-Determinant. We consider using this rank approximation for subspace clustering application. Our framework can model different kinds of errors and noise. Effective optimization strategy is developed with theoretical guarantee to converge to a stationary point. The proposed method gives promising results on face clustering and motion segmentation tasks compared to the state-of-the-art subspace clustering algorithms.