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 _1-norms are used to measure rank and sparsity. However, the use of nuclear and _1-norms leads to an overpenalized problem and only approximates the original problem. In this paper, we propose two _0 quasi-norm-based regularizations. First, this paper presents regularization based on multivariate generalization of minimax-concave penalty (GMC-LRSSC), which contains the global minimizers of a _0 quasi-norm regularized objective. Afterward, we introduce the Schatten-0 (S0) and _0-regularized objective and approximate the proximal map of the joint solution using a proximal average method (S0/_0-LRSSC). The resulting nonconvex optimization problems are solved using an 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/_0-LRSSC when compared to state-of-the-art methods.

中文翻译：

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$\ell_0$ - 激励的低秩稀疏子空间聚类


在许多应用中，高维数据点可以用低维子空间很好地表示。为了识别子空间，捕获数据的全局和局部结构非常重要，这是通过对数据表示矩阵施加低秩和稀疏性约束来实现的。在低秩稀疏子空间聚类（LRSSC）中，核范数和_1范数用于测量秩和稀疏性。然而，使用核和 _1-范数会导致过度惩罚问题，并且只能近似原始问题。在本文中，我们提出了两种基于 _0 准范数的正则化。首先，本文提出了基于极小最大凹罚分的多元广义化（GMC-LRSSC）的正则化，其中包含_0准范数正则化目标的全局最小化器。之后，我们引入 Schatten-0 (S0) 和 _0 正则化目标，并使用近端平均方法 (S0/_0-LRSSC) 近似联合解的近端图。由此产生的非凸优化问题可以使用乘法器的交替方向方法来解决，并建立两种算法的收敛条件。与最先进的方法相比，在合成数据集和四个真实数据集上获得的结果显示了 GMC-LRSSC 和 S0/_0-LRSSC 的有效性。