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The subspace segmentation problem is addressed in this paper by effectively constructing an \emph{exactly block-diagonal} sample affinity matrix. The block-diagonal structure is heavily desired for accurate sample clustering but is rather difficult to obtain. Most current state-of-the-art subspace segmentation methods (such as SSC~\cite{elhamifar2012sparse} and LRR~\cite{LiuLYSYM13}) resort to alternative structural priors (such as sparseness and low-rankness) to construct the affinity matrix. In this work, we directly pursue the block-diagonal structure by proposing a graph Laplacian constraint based formulation, and then develop an efficient stochastic subgradient algorithm for optimization. Moreover, two new subspace segmentation methods, the block-diagonal SSC and LRR, are devised in this work. To the best of our knowledge, this is the first research attempt to explicitly pursue such a block-diagonal structure. Extensive experiments on face clustering, motion segmentation and graph construction for semi-supervised learning clearly demonstrate the superiority of our novelly proposed subspace segmentation methods.

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