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: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. 18901v1 Announce Type: cross Abstract: In physics-informed machine learning, a target function $u^*$ is learned from noisy value observations $y_i=u^*(x_i)+ \varepsilon_i$, together with differential information, given either by noisy observations $d_j=(Du^*)(z_j)+\xi_j$ or by a known physical constraint $Du^*=v$.
By Luc Brogat-Motte, Joachim Bona-Pellissier, Giacomo Meanti, Lorenzo Rosasco
The paper introduces Shared Gaussianization (SG), a characteristic‑function test that evaluates Gaussianity of the average of two normalized views scaled by an independent χ_d radius. SG detects both misalignment and non‑uniformity in contrastive learning, vanishing exactly at aligned, uniform minimizers of population InfoNCE, and provides a sharp, dimension‑free bound on InfoNCE excess. The study also explores how SG behaves under different kernels, alignment weights, batch sizes, and latent‑variable models, revealing optimization gaps and the impact of nuisance channels.
By Ruoyu Zhao, Yuting Chen, Jinheng Zhang, Zhehao Zou, Tong Che
arXiv:2501. 10870v2 Announce Type: replace-cross Abstract: The principal objective of this work is twofold within nonparametric regression settings: (1) to establish the minimax optimal convergence rates for fixed-bandwidth Gaussian kernel spectral algorithms when the true regression function resides in a Sobolev space, and (2) to apply Gaussian spectral algorithms for achieving robust and adaptive transfer learning under concept shift.
By Haotian Lin, Matthew Reimherr
arXiv:2610.27022v1 Announce Type: cross
Abstract: We study Gaussian kernel approximation and regularized logistic classification on a connected compact smooth embedded manifold $X\subset\mathbb R^d$...
By Guanhang Lei, Lei Shi, Xiaoming Yuan
arXiv:2609. 03129v1 Announce Type: cross Abstract: Several classical machine-learning methods, such as KRRs and SVRs, are both computationally and analytically tractable since their estimators either admit closed-form expressions or are obtained by minimizing convex training objectives; neither feature is generally available for deep neural networks.
By Ruiyang Hong, Hrad Ghoukasian, Anastasis Kratsios
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:2608.29265v1 Announce Type: cross
Abstract: Kernel density estimation (KDE) is one of the most fundamental statistical estimators of density functions. Its direct implementation on a dataset of...
By Xie Wang, Nicolas Langren\'e, Wen Chen
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:2606. 20183v1 Announce Type: new Abstract: Recent quantum vision models-quantum vision transformers and quantum convolutional networks-report two striking but unexplained empirical phenomena: (i) ansatze with more, or more uniformly distributed, entanglement generalize better, and (ii) injecting quantum noise can improve test accuracy rather than degrade it.
By Jian Xu, Delu Zeng, John Paisley, Qibin Zhao
arXiv:2605. 31152v2 Announce Type: replace-cross Abstract: This paper studies how efficiently deep ReLU neural networks can approximate and learn smooth functions.
By Yunfei Yang, Jun Fan