Let X be a data matrix of rank ρ, whose rows represent n points in d-dimensional space. The linear support vector machine constructs a hyperplane separator that maximizes the 1-norm soft margin. We develop a new oblivious dimension reduction technique that is precomputed and can be applied to any input matrix X. We prove that, with high probability, the margin and minimum enclosing ball in the feature space are preserved to within -relative error, ensuring comparable generalization as in the

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... ginal space in the case of classification. For regression, we show that the margin is preserved to -relative error with high probability. We present extensive experiments with real and synthetic data to support our theory. A short version of this article appeared in the 16th International Conference on Artificial Intelligence and Statistics (AISTATS 2013) [Paul et al. 2013] . Note that the short version of our article [Paul et al. 2013] does not include the details of the proofs, comparison of random projections with principal component analysis, extension of random projections for SVM regression in terms of both theory and experiments, and experiments with fast SVM solver on RCV1 and Hapmap-HGDP datasets.