Sparse subspace clustering via smoothed ℓ_<i>p</i> minimization

In this letter, we formulate sparse subspace clustering as a smoothed ℓp (0 ˂ p ˂ 1) minimization problem (SSC-SLp) and present a unified formulation for different practical clustering problems by introducing a new pseudo norm. Generally, the use of ℓp (0 ˂ p ˂ 1) norm approximating the ℓ0 one can lead to a more effective approximation than the ℓp norm, while the ℓp-regularization also causes the objective function to be non-convex and non-smooth. Besides, better adapting to the property of data representing real problems, the objective function is usually constrained by multiple factors (such as spatial distribution of data and errors). In view of this, we propose a computationally efficient method for solving the multi-constrained non-smooth ℓp minimization problem, which smooths the ℓp norm and minimizes the objective function by alternately updating a block (or a variable) and its weight. In addition, the convergence of the proposed algorithm is theoretically proven. Extensive experimental results on real datasets demonstrate the effectiveness of the proposed method