Face recognition has been widely studied due to its importance in various applications. However, the case that both training images and testing images are corrupted is not well solved. To address such a problem, this paper proposes a semisupervised learning algorithm for robust face recognition. In particular, we consider three items in the proposed formulation. First, a low-rank and sparse representation for face recognition is required to handle the possible contamination of the whole data. Second, a classwise block-diagonal structure of the learned representation is expected to promote discrimination among different classes. With the structure regularization, we make the samples from different classes be reconstructed with different bases as much as possible. Third, a compact and discriminative dictionary should be learnt to handle the problem of corrupted data. Extensive experiments on three public databases are performed to validate the effectiveness of our approach. The strong identification capability of representation with block-diagonal structure is verified.