ℓ0-Motivated Low-Rank Sparse Subspace Clustering

In many applications, high-dimensional data points can be well represented by low-dimensional subspaces. To identify the subspaces, it is important to capture a global and local structure of the data which is achieved by imposing low-rank and sparseness constraints on the data representation matrix. In low-rank sparse subspace clustering (LRSSC), nuclear and ℓ1 norms are used to measure rank and sparsity. However, the use of nuclear and ℓ1 norms leads to an overpenalized problem and only approximates the original problem. In this paper, we propose two ℓ0 quasi-norm based regularizations. First, the paper presents regularization based on multivariate generalization of minimax-concave penalty (GMC-LRSSC), which contains the global minimizers of ℓ0 quasi-norm regularized objective. Afterward, we introduce the Schatten-0 (S0) and ℓ0 regularized objective and approximate the proximal map of the joint solution using a proximal average method (S0/ℓ0-LRSSC). The resulting nonconvex optimization problems are solved using alternating direction method of multipliers with established convergence conditions of both algorithms. Results obtained on synthetic and four real-world datasets show the effectiveness of GMC-LRSSC and S0/ℓ0-LRSSC when compared to state-of-the-art methods.