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 0-minimization problem with 1-norm. Although the use of 1-norm instead of the 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 0-minimization than the traditional 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. INDEX TERMS Sparse subspace clustering, nonconvex approximation, ADMM, reweighted algorithms.

Sparse subspace clustering algorithm with non-convex constraints

“…The Alternating Direction method of Multipliers (ADMM) [18] is used to solve the sparse subspace clustering algorithm based on non-convex φ  energy functional.…”

“…Least Squares Regression (LSR) [16] uses Fnorm to minimize the representation coefficient to obtain the representation coefficient matrix, which can group highly correlated data together.Re-weighted Sparse Subspace Clustering (RSSC) [17] adds weights to the representation coefficients of the 1 l -norm constraint to reconstruct the representation coefficient matrix, which is closer to the 0 l -norm than the traditional 1 l -norm. Non-convex applications of sparse subspace clustering [18] performs clustering by solving the representation coefficients using , q l  to replace the 0 l -norm constraint objective function. Although the 1 l -norm is used as the constraint penalty term to make the objective function convex, it can not accurately describe the sparsity of coefficient matrix.…”

Nonconvex Sparse Representation With Slowly Vanishing Gradient Regularizers

“…Hence, we might be interested in finding a good nonconvex surrogate for such general nonconvex problems. Prior works support the superiority of nonconvex sparsity-promoting surrogates [9], [12], [26]- [31].…”

Robust Multiview Subspace Clustering of Images via Tighter Rank Approximation

“…The most representative ones are sparse subspace clustering [10] (SSC, pursuing a sparse coefficient matrix) and low-rank subspace clustering [11] (LRR, pursuing a low-rank coefficient matrix). Thereafter, many extensions have been developed to improve these methods, such as sparse subspace clustering algorithm based on a nonconvex modeling formulation [12] and sketchbased subspace clustering [13]. FIGURE 1: The flowchart of our method.…”