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

Abstract

We introduce the Yat kernel $$k_{b,\epsilon }(\mathbf{w},\mathbf{x})=\frac{({\mathbf{w}}^{\top }\mathbf{x}+b{)}^2}{\Vert \mathbf{x}-\mathbf{w}{\Vert }^2+\epsilon },b\ge 0,\epsilon >0,$$ a rational hidden-unit primitive whose units are Mercer sections over a shared input/weight space. For $b\ge 0$ the kernel is PSD; for $b>0$ it dominates a scaled inverse-multiquadric (IMQ) in the Loewner order, yielding fixed-kernel universality, characteristicness, and strict positive definiteness on every compact domain. The polynomial numerator opens nonradial alignment channels absent from finite IMQ expansions, witnessed by the directional far-field trace $T_{\infty }{g}_{\epsilon }(\cdot ;\mathbf{w},b)(\mathbf{u})=({\mathbf{u}}^{\top }\mathbf{w}{)}^2$. Algebraically, a second finite difference in the bias recovers any IMQ atom from three positive-bias Yat atoms exactly, sharp at three atoms in every dimension at exact pointwise equality. A trained shared-$(b,\epsilon )$ Yat layer is therefore a finite learned-center expansion in a fixed universal characteristic RKHS, with closed-form norm $\hat{\mathbf{I}}{\pm }^{\top }\mathbf{K}\hat{\mathbf{I}}\pm$ and explicit diagonal $(\Vert \mathbf{x}{\Vert }^2+b{)}^2/\epsilon$ driving a Rademacher generalization bound.