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We propose $\ell_1$ norm regularized quadratic surface support vector machinemodels for binary classification in supervised learning. We establish theirdesired theoretical properties, including the existence and uniqueness of theoptimal solution, reduction to the standard SVMs over (almost) linearlyseparable data sets, and detection of true sparsity pattern over (almost)quadratically separable data sets if the penalty parameter of $\ell_1$ norm islarge enough. We also demonstrate their promising practical efficiency byconducting various numerical experiments on both synthetic and publiclyavailable benchmark data sets.

Quadratic Surface Support Vector Machine with L1 Norm Regularization