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Linear subspace methods that provide sufficient reconstruction of the data, such as PCA, offer an efficient way of dealing with missing pixels, outliers, and occlusions that often appear in the visual data. Discriminative methods, such as LDA, which, on the other hand, are better suited for classification tasks, are highly sensitive to corrupted data. We present a theoretical framework for achieving the best of both types of methods: an approach that combines the discrimination power of discriminative methods with the reconstruction property of reconstructive methods which enables one to work on subsets of pixels in images to efficiently detect and reject the outliers. The proposed approach is therefore capable of robust classification with a high-breakdown point. We also show that subspace methods, such as CCA, which are used for solving regression tasks, can be treated in a similar manner. The theoretical results are demonstrated on several computer vision tasks showing that the proposed approach significantly outperforms the standard discriminative methods in the case of missing pixels and images containing occlusions and outliers.