We formulate a novel regularized risk minimization problem for learning in reproducing kernel Kre{ı̆}n spaces and show that the strong representer theorem applies to it. As a result of the latter, the learning problem can be expressed as the minimization of a quadratic form over a hypersphere of constant radius... We present an algorithm that can find a globally optimal solution to this non-convex optimization problem in time cubic in the number of instances. Moreover, we derive the gradient of the solution with respect to its hyperparameters and, in this way, provide means for efficient hyperparameter tuning. The approach comes with a generalization bound expressed in terms of the Rademacher complexity of the corresponding hypothesis space. The major advantage over standard kernel methods is the ability to learn with various domain specific similarity measures for which positive definiteness does not hold or is difficult to establish. The approach is evaluated empirically using indefinite kernels defined on structured as well as vectorial data. The empirical results demonstrate a superior performance of our approach over the state-of-the-art baselines. (read more)