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

arXiv:2605.03262v1 Announce Type: new Abstract: We introduce the Yat kernel $k_{b,arepsilon}(mathbf{w},mathbf{x})=frac{(mathbf{w}^opmathbf{x}+b)^2}{|mathbf{x}-mathbf{w}|^2+arepsilon},qquad bge 0, arepsilon>0, a rational hidden-unit primitive whose units are Mercer sections over a shared input/weight space. For bge 0 the kernel is PSD; for b>0 it dominates a scaled inverse-multiquadric (IMQ) in the Loewner order, yielding fixed-kernel universality, characteristicness, and strict positive definiteness on every compact domain. The polynomial numerator opens nonradial alignment channels absent from finite IMQ expansions, witnessed by the directional far-field trace T_infty g_arepsilon(dot;mathbf{w},b)(mathbf{u})=(mathbf{u}^opmathbf{w})^2. Algebraically, a second finite difference in the bias recovers any IMQ atom from three positive-bias Yat atoms exactly, sharp at three atoms in every dimension at exact pointwise equality. A trained shared-(b,arepsilon) Yat layer is therefore a finite learned-center expansion in a fixed universal characteristic RKHS, with closed-form norm oldsymbol{alpha}^opmathbf{K}oldsymbol{alpha} and explicit diagonal (|mathbf{x}|^2+b)^2/arepsilon$ driving a Rademacher generalization bound.

Source: arXiv cs.LG | 2026-05-06