${\ell }_0$-Motivated Low-Rank Sparse Subspace Clustering

Abstract

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 ${\ell }_1$ norms are used to measure rank and sparsity. However, the use of nuclear and ${\ell }_1$ norms leads to an overpenalized problem and only approximates the original problem. In this paper, we propose two ${\ell }_0$ 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 ${\ell }_0$ quasi-norm regularized objective. Afterward, we introduce the Schatten-0 ($S_0$) and ${\ell }_0$ regularized objective and approximate the proximal map of the joint solution using a proximal average method ($S_0/{\ell }_0$-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 $S_0/{\ell }_0$-LRSSC when compared to state-of-the-art methods.

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