A Nonconvex Implementation of Sparse Subspace Clustering: Algorithm and Convergence Analysis

Abstract

Subspace clustering has been widely applied to detect meaningful clusters in high-dimensional data spaces. And the sparse subspace clustering (SSC) obtains superior clustering performance by solving a relaxed $\ell _{0}$ -minimization problem with $\ell _{1}$ -norm. Although the use of $\ell _{1}$ -norm instead of the $\ell _{0}$ one can make the object function convex, it causes large errors on large coefficients in some cases. In this paper, we study the sparse subspace clustering algorithm based on a nonconvex modeling formulation. Specifically, we introduce a nonconvex pseudo-norm that makes a better approximation to the $\ell _{0}$ -minimization than the traditional $\ell _{1}$ -minimization framework and consequently finds a better affinity matrix. However, this formulation makes the optimization task challenging due to that the traditional alternating direction method of multipliers (ADMM) encounters troubles in solving the nonconvex subproblems. In view of this, the reweighted techniques are employed in making these subproblems convex and easily solvable. We provide several guarantees to derive the convergence results, which proves that the nonconvex algorithm is globally convergent to a critical point. Experiments on two real-world problems of motion segmentation and face clustering show that our method outperforms state-of-the-art techniques.