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In this paper, application of sparse representation (factorization) of signals over an overcomplete basis (dictionary) for signal classification is discussed. Searching for the sparse representation of a signal over an overcomplete dictionary is achieved by optimizing an objective function that includes two terms: one that measures the signal reconstruction error and another that measures the sparsity. This objective function works well in applications where signals need to be reconstructed, like coding and denoising. On the other hand, discriminative methods, such as linear discriminative analysis (LDA), are better suited for classification tasks. However, discriminative methods are usually sensitive to corruption in signals due to lacking crucial properties for signal reconstruction. In this paper, we present a theoretical framework for signal classification with sparse representation. The approach combines the discrimination power of the discriminative methods with the reconstruction property and the sparsity of the sparse representation that enables one to deal with signal corruptions: noise, missing data and outliers. The proposed approach is therefore capable of robust classification with a sparse representation of signals. The theoretical results are demonstrated with signal classification tasks, showing that the proposed approach outperforms the standard discriminative methods and the standard sparse representation in the case of corrupted signals.