ℓ_0-Motivated Low-Rank Sparse Subspace Clustering
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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 ℓ_1 norms are used to measure rank and sparsity. However, the use of nuclear and ℓ_1 norms leads to an overpenalized problem and only approximates the original problem. In this paper, we propose two ℓ_0 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 ℓ_0 quasi-norm regularized objective. Afterward, we introduce the Schatten-0 (S_0) and ℓ_0 regularized objective and approximate the proximal map of the joint solution using a proximal average method (S_0/ℓ_0-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 S_0/ℓ_0-LRSSC when compared to state-of-the-art methods.
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