  In this paper, we study the asymptotic properties of regularized least
squares with indefinite kernels in reproducing kernel Krein spaces (RKKS). 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
approximation analysis feasible in RKKS. Regarding to the original regularizer
induced by the indefinite inner product, 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.
