Full-rank orthonormal bases for conditionally positive definite kernel-based spaces

In kernel-based approximation, it is well-known that the direct approach to interpolation is prone to ill-conditioning of the interpolation matrix. One simple idea is to use other better-conditioned bases that span the same space of the translated kernels i.e. their associated native space. Pazouki and Schaback (2011) tracked this issue by investigating different factorization of the interpolation matrix in order to build stable and orthonormal bases for the corresponding native space of the positive definite kernels. In this paper, we work with the reproducing kernel K for the associated native Hilbert space NΦ(Ω) corresponding to a conditionally positive definite kernel Φ on the nonempty set Ω. We give a well-organized matrix formulation of the kernel matrix K by constructing the matrices corresponding to cardinal basis from monomials. Then, we present two possible ways to find full-rank data-dependent orthonormal bases that are discretely ℓ2 and NΦ-orthonormal. The first approach is given by the factorization of the kernel matrix K and the next one is based on the eigenpairs approximation of the linear operator associated with the reproducing kernel K given by Mercer's theorem. In the sequel, we employ the truncated singular value decomposition technique to find an optimal low-rank basis with the coefficient matrix whose rank is less than that of the original matrix. Special attention is also given to error analysis, duality, and stability. Some numerical experiments are also provided.