arXiv Machine Learning
Jun 11

Bernstein-Schur Kernels: Random Features by Sketched Modulation and Radial Randomization

arXiv:2606. 11255v1 Announce Type: new Abstract: Bernstein--Schur kernels are products of a finite-feature kernel (one with an explicit finite-dimensional feature map) and a completely monotone shift-invariant kernel: nonstationary kernels that fall between the shift-invariant and dot-product templates random features usually exploit, so in general neither Bochner sampling nor polynomial sketching applies to the full kernel directly.

By Taha Bouhsine
arXiv AI
Sep 4

Spectral Convergence of Random Feature Method in Multiple Dimensions

The paper proves spectral convergence of the random feature method (RFM) for multidimensional targets across various function classes, providing high‑probability approximation estimates that hold simultaneously for all admissible error norms. It extends these results to strong‑ and weak‑form RFM discretizations, yielding convergence guarantees for multidimensional second‑order elliptic boundary value and eigenvalue problems. Additionally, it demonstrates super‑exponential singular‑value decay for Fourier features and exponential decay for tanh features, while establishing corresponding condition‑number lower bounds, highlighting a trade‑off between accuracy and ill‑conditioning.

By Pingbing Ming, Hao Yu
arXiv Machine Learning
Jul 13

A Fourier analytique approach to Gaussian mixture learning

arXiv:2004. 05813v3 Announce Type: replace-cross Abstract: Suppose that we are given independent, identically distributed random samples $x_1,\cdots,x_n$ from a mixture at most $k$ many $d$-dimensional spherical Gaussian distributions $\mu_1,\cdots,\mu_{k_0}$ of identical and known variance $\sigma^2$ in each coordinate, such that the minimum $\ell^2$ distance between two distinct centers $y_l$ and $y_j$ is greater than $2\Delta\sigma \min\{\sqrt{d},\sqrt k\}$, where $\Delta>C_0$, and $C_0$ is a sufficiently large universal constant.

By Somnath Chakraborty, Hariharan Narayanan