提出一种新型核函数，可实现通用特征空间并提升模型表达能力。
A Universal Reproducing Kernel Hilbert Space from Polynomial Alignment and IMQ Distance
- 设计含多项式分子的有理核，支持非径向对齐机制
- 在任意紧凑域上保证核的特征性与严格正定性
- 适用于需要高表达力的机器学习任务，如深度网络优化
我们引入一种新型核函数 $k_{b, u}(oldsymbol{w},oldsymbol{x}) = \frac{(\boldsymbol{w}^\top\boldsymbol{x}+b)^2}{\|\boldsymbol{x}-\boldsymbol{w}\|^2+\varepsilon}$，其中 $b\ge 0$，$\varepsilon>0$。该核为共享输入/权重空间上的Mercer分量构成的有理隐藏单元。当 $b\ge 0$ 时核为半正定；当 $b>0$ 时其在Loewner序中支配缩放后的逆多二次（IMQ）核，从而在任意紧致域上实现固定核的通用性、特征性及严格正定性。多项式分子引入了有限IMQ展开所不具备的非径向对齐通道，体现于方向远场迹 $T_\infty g_\varepsilon(\cdot;\boldsymbol{w},b)(\boldsymbol{u})=(\boldsymbol{u}^\top\boldsymbol{w})^2$。代数上，通过偏置的二阶差分可精确从三个正偏置的Yat原子恢复任一IMQ原子，且在每个维度下三点点对点等价时即精确成立。因此，训练共享 $(b,\varepsilon)$ 的Yat层等价于在固定通用特征型希尔伯特空间中的有限学习中心展开，具有闭式范数 $\boldsymbol{\alpha}^\top\mathbf{K}\boldsymbol{\alpha}$ 及显式对角项 $(\|\boldsymbol{x}\|^2+b)^2/\varepsilon$，驱动一个Rademacher泛化界。
原文摘要 · Abstract (English)
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.
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