标题： 岭核平均与一致逼近 显示英文标题

标题： Ridge Kernel Averaging and Uniform Approximation

摘要： 我们开发了一个由参数脊核生成的函数类框架：一维核与仿射投影的组合，并在参数测度上取平均。 由此产生的核是正定的，所得积分类与它的再生核希尔伯特空间等距重合。 我们通过变化测度得到的所有核表征为脊原子的锥包的均匀闭包，给出了一个精确的普遍性准则；逐片的多项式与非多项式二分法决定了表达能力。 然后我们分析了激活函数可能不定但具有正定均值的随机核网络。 对于这些网络，我们证明了蒙特卡洛速率，均方误差为1除以$N$的阶，并在紧集上证明了高概率的一致界，而无需路径上的正定性。 显示英文摘要

摘要： We develop a framework for function classes generated by parametric ridge kernels: one-dimensional kernels composed with affine projections and averaged over a parameter measure. The induced kernels are positive definite, and the resulting integral class coincides isometrically with its reproducing kernel Hilbert space. We characterize all kernels obtainable by varying the measure as the uniform closure of the conic hull of ridge atoms, giving a sharp universality criterion; a slice-wise polynomial versus non-polynomial dichotomy governs expressivity. We then analyze random-kernel networks whose activations may be indefinite but have a positive-definite mean. For these networks we prove a Monte Carlo rate with mean-squared error of order one over $N$ and a high-probability uniform bound on compact sets, without requiring pathwise positive definiteness.