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The development of cancer diagnosis models and cancer discovery from DNA microarray data are of great interest in bioinformatics and medicine. In pattern recognition and machine learning, a classification problem refers to finding an algorithm for assigning a given input data into one of several categories. Many natural signals are sparse or compressible in the sense that they have short representations when expressed in a suitable basis. Motivated by the recent successful algorithm developments for sparse signal recovery, we apply the selective nature of sparse representation to perform the above mentioned classification. In order to find such sparse representation we implement an ℓ1-minimization algorithm. This methodology overcomes the lack of robustness with respect to outliers. In contrast to other classification algorithms, no model selection dependency is involved. The minimization algorithm is a convex relaxation-like that has been proven to efficiently recover sparse signals. To study its performance, the proposed method is applied to six tumor gene expression datasets and numerically compared with various support vector machine methods (SVM). The numerical results show that the ℓ1-minimization algorithm proposed performs at least comparably and often better than SVMs.