Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2605.03262 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866917460857323520 |
|---|---|
| author | Bouhsine, Taha |
| author_facet | Bouhsine, Taha |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_03262 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Universal Reproducing Kernel Hilbert Space from Polynomial Alignment and IMQ Distance Bouhsine, Taha Machine Learning 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. |
| title | A Universal Reproducing Kernel Hilbert Space from Polynomial Alignment and IMQ Distance |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2605.03262 |