A Universal Reproducing Kernel Hilbert Space from Polynomial Alignment and IMQ Distance

Abstract

Inverse-multiquadric (IMQ) kernel sections decay at infinity. Can polynomial alignment add persistent directional responses while preserving universal approximation? The \yat{} (Yat) kernel answers this question by multiplying IMQ distance by a biased squared inner product. With shared positive bias and regularization, its RKHS continuously contains the IMQ RKHS and is universal on every compact domain. Alignment strictly enlarges the global function space: Yat sections retain a quadratic directional trace at infinity. We quantify a second distinction through high-dimensional activation covariance. For a fixed number of orthogonal unit centers and fixed kernel parameters, isotropic Gaussian inputs yield a rank-one rescaled limiting covariance for IMQ and a full-rank limit for Yat. On the radius-$\sqrt d$ sphere, both limits are full rank, isolating shared input-norm fluctuations as the Gaussian collapse mechanism. The connection is also constructive: three positive-bias Yat atoms recover one IMQ section exactly, and three are necessary at nonzero centers when regularization is shared and bias varies across atoms. These results identify how polynomial alignment preserves radial approximation while adding directional structure.