  We propose $\ell_1$ norm regularized quadratic surface support vector machine
models for binary classification in supervised learning. We establish their
desired theoretical properties, including the existence and uniqueness of the
optimal solution, reduction to the standard SVMs over (almost) linearly
separable data sets, and detection of true sparsity pattern over (almost)
quadratically separable data sets if the penalty parameter of $\ell_1$ norm is
large enough. We also demonstrate their promising practical efficiency by
conducting various numerical experiments on both synthetic and publicly
available benchmark data sets.
