🤖 AI Summary
This work addresses the challenge that existing reproducing kernel Hilbert space (RKHS) kernels struggle to simultaneously achieve universality, characteristicness, and strict positive definiteness. To resolve this, the paper introduces a novel kernel, termed the Yat kernel, which innovatively incorporates a polynomial alignment mechanism into inverse multiquadric (IMQ)-type kernels. By combining polynomial numerators with IMQ-based distance denominators, the Yat kernel constructs a positive definite kernel sensitive to non-radial directions within shared input or weight spaces. Theoretical analysis demonstrates that the Yat kernel attains universality, characteristicness, and strict positive definiteness over any compact domain, admits finite center expansions, and—through its three bias-equipped atomic components—exactly reconstructs arbitrary IMQ atoms in a dimension-independent manner. Leveraging Mercer’s theorem, Loewner order comparisons, and Rademacher complexity analysis, the study derives explicit generalization error bounds, confirming the model’s strong expressivity and learning stability.
📝 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 $\boldsymbolα^\top\mathbf{K}\boldsymbolα$ and explicit diagonal $(\|\mathbf{x}\|^2+b)^2/\varepsilon$ driving a Rademacher generalization bound.