Analysis of Least Squares Regularized Regression in Reproducing Kernel Krein Spaces

In this paper, we study the asymptotical properties of least squares regularized regression with indefinite kernels in reproducing kernel Kre\u{\i}n spaces (RKKS). The classical approximation analysis cannot be directly applied to study its asymptotical behavior under the framework of learning theory as this problem is in essence non-convex and outputs stationary points. By introducing a bounded hyper-sphere constraint to such non-convex regularized risk minimization problem, we theoretically demonstrate that this problem has a globally optimal solution with a closed form on the sphere, which makes our approximation analysis feasible in RKKS. Accordingly, we modify traditional error decomposition techniques, prove convergence results for the introduced hypothesis error based on matrix perturbation theory, and derive learning rates of such regularized regression problem in RKKS. Under some conditions, the derived learning rates in RKKS are the same as that in reproducing kernel Hilbert spaces (RKHS), which is actually the first work on approximation analysis of regularized learning algorithms in RKKS.