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:2409. 08469v4 Announce Type: replace-cross Abstract: We provide finite-particle convergence rates for the Stein Variational Gradient Descent (SVGD) algorithm in the Kernelized Stein Discrepancy ($\mathsf{KSD}$) and Wasserstein-2 metrics.
By Sayan Banerjee, Krishnakumar Balasubramanian, Promit Ghosal
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:2610.33286v1 Announce Type: cross
Abstract: Given a random probability distribution \(F\), whose realizations are defined on \(\mathbb R^d\), and a kernel \(k\), our aim is to analyze the prope...
By Amparo Ba\'illo, Mart\'in S\'anchez-Signorini
arXiv:2610.09731v1 Announce Type: cross
Abstract: Kernel methods are among the most powerful tools in machine learning and statistics, with a large number of successful applications. Their immense su...
By Jose Cribeiro-Ramallo, Florian Kalinke, Zolt\'an Szab\'o
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
The paper studies random projections (RP) for spectral algorithms, providing a comprehensive analysis of their generalization performance under broad conditions. By exploiting the RKHS embedding property and operator similarity, it establishes optimal learning rates in Sobolev norms that match minimax lower bounds up to logarithmic factors. For common RP implementations—randomized sketches and Nyström sub-sampling—the required projection dimension scales with the effective dimension, reducing computational cost while preserving statistical efficiency.
By Caixing Wang, Yue Wang
arXiv:2410. 23212v3 Announce Type: replace-cross Abstract: In graph-based data analysis, $k$-nearest neighbor ($k$NN) graphs are widely used due to their adaptivity to local data densities.
By Xiuyuan Cheng, Yixuan Tan, Nan Wu
arXiv:2609.15179v1 Announce Type: cross
Abstract: The Gaussian kernel is a widely used similarity measure underlying kernel methods such as kernel PCA and spectral clustering, but computing Gaussian...
By Soumik Dutta, Kunal Dutta
arXiv:2607. 24235v1 Announce Type: cross Abstract: Over the past 20 years, kernel discrepancies have been leveraged as a highly powerful tool for quantifying the disagreement of distributions, with numerous successful applications in two-sample, goodness-of-fit, and independence testing, among others.
By Jose Cribeiro-Ramallo, Florian Kalinke, Zolt\'an Szab\'o
arXiv:2609. 18577v1 Announce Type: new Abstract: We consider the problem of estimating the trace of an implicit matrix $\mathbf{A} \in \mathbb{R}^{d^p\times d^p}$ that can only be accessed through matrix-vector products queries.
By Mohammad Azhar Khan, Rameshwar Pratap, Amit Sharma
\texttt{TensorSketch} by~\cite{pham2013fast,kar2012random} provides efficient sketching algorithms for high-dimensional polynomial kernels $\vec{x}^{\otimes p} \in \R^{d^p}$. \cite{kar2012random} uses dense Johnson-Lindenstrauss (JL)-type projections with computational cost $O(pDd)$, where $D$ denotes the sketch dimension, whereas~\cite{pham2013fast} extends the sparse \texttt{CountSketch}~\citep{count_sketch} algorithm, yielding a faster algorithm for high-dimensional sparse inputs with running time $O\big(p(\nnz{\vec{x}} + D \log D)\big)$.