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

TL;DR

This paper introduces a new universal kernel called the Yat kernel, which combines polynomial and IMQ features to enable fixed-kernel universality and explicit learned-center expansions.

Contribution

The paper presents the Yat kernel, a novel positive semi-definite kernel with fixed universality, characteristicness, and explicit finite learned-center expansions.

Findings

Yat kernel dominates a scaled IMQ, ensuring universality.

Finite differences recover IMQ atoms exactly in the Yat kernel.

Shared-$(b,psilon)$ Yat layer acts as a finite learned-center expansion with explicit norm.

Abstract

We introduce the Yat kernel $$k_{b,\varepsilon}(\mathbf{w},\mathbf{x})=\frac{(\mathbf{w}^\top\mathbf{x}+b)^2}{\|\mathbf{x}-\mathbf{w}\|^2+\varepsilon},\qquad b\ge 0,\ \varepsilon>0,$$ 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 $T_\infty g_\varepsilon(\cdot;\mathbf{w},b)(\mathbf{u})=(\mathbf{u}^\top\mathbf{w})^2$. 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-$(b,\varepsilon)$ Yat layer is therefore a finite learned-center expansion in a fixed universal characteristic RKHS, with closed-form norm $\boldsymbol{\alpha}^\top\mathbf{K}\boldsymbol{\alpha}$ and explicit diagonal $(\|\mathbf{x}\|^2+b)^2/\varepsilon$ driving a Rademacher generalization bound.