Sparse subspace clustering via nonconvex approximation

Abstract

Among existing clustering methods, sparse subspace clustering (SSC) obtains superior clustering performance in grouping data points from a union of subspaces by solving a relaxed $\ell _{0}$-minimization problem by $\ell _{1}$-norm. The use of $\ell _{1}$-norm instead of the $\ell _{0}$ one can make the object function convex, while it also causes large errors on large coefficients in some cases. In this work, we propose using the nonconvex approximation to replace $\ell _{0}$-norm for SSC, termed as SSC via nonconvex approximation (SSCNA), and develop a novel clustering algorithm with the enhanced sparsity based on the Alternating Direction Method of Multipliers. We further prove that the proposed nonconvex approximation is closer to $\ell _{0}$-norm than the $\ell _{1}$ one and is bounded by $\ell _{0}$-norm. Numerical studies show that the proposed nonconvex approximation helps to improve clustering performance. We also theoretically verify the convergence of the proposed algorithm with a three-variable objective function. Extensive experiments on four benchmark datasets demonstrate the effectiveness of the proposed method.