Sparse graphs built by sparse representation has been demonstrated to be effective in clustering high-dimensional data. Albeit the compelling empirical performance, the vanilla sparse graph ignores the geometric information of the data by performing sparse representation for each datum separately. In order to obtain a sparse graph aligned with the local geometric structure of data, we propose a novel Support Regularized Sparse Graph, abbreviated as SRSG, for data clustering. SRSG encourages local smoothness on the neighborhoods of nearby data points by a well-defined support regularization term. We propose a fast proximal gradient descent method to solve the non-convex optimization problem of SRSG with the convergence matching the Nesterov's optimal convergence rate of first-order methods on smooth and convex objective function with Lipschitz continuous gradient. Extensive experimental results on various real data sets demonstrate the superiority of SRSG over other competing clustering methods.

翻译：通过稀疏表示构建的稀疏图已被证明在聚类高维数据方面具有显著效果。尽管经验性能优异，传统稀疏图因对每个数据点单独进行稀疏表示而忽略了数据的几何信息。为获得与数据局部几何结构对齐的稀疏图，本文提出一种新颖的支持正则化稀疏图（简称SRSG）用于数据聚类。SRSG通过定义完善的支持正则化项，在邻近数据点的邻域内促进局部平滑性。我们提出一种快速近端梯度下降法来求解SRSG的非凸优化问题，其收敛速度匹配了Nesterov关于具有Lipschitz连续梯度的光滑凸目标函数的一阶方法最优收敛率。在多个真实数据集上的大量实验结果证明了SRSG相较于其他聚类方法的优越性。