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.

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