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:2504.18184v5 Announce Type: replace
Abstract: We consider a class of statistical inverse problems involving the estimation of a regression operator from a Polish space to a separable Hilbert sp...
By Jia-Qi Yang, Lei Shi
arXiv:2606. 27298v1 Announce Type: cross Abstract: We study the fundamental problem of learning a high-dimensional Gaussian truncated to an unknown halfspace.
By Haitong Liu, Deepak Narayanan Sridharan, David Steurer, Manuel Wiedmer
The paper introduces a Projected Riemannian Gradient Descent (RGD) algorithm for computing the Bures‑Wasserstein barycenter of positive definite matrices, achieving dimension‑independent linear convergence at unit step size. It resolves a previous dichotomy by showing that clipping eigenvalues to a fixed interval yields a closed‑form, non‑expansive projection in the BW metric, allowing the algorithm to match the empirical speed of unit‑step RGD while maintaining theoretical guarantees. The method also extends to the invariant matrix projection problem, providing a unified dimension‑independent analysis.
The paper studies operator learning on function spaces using encoder–decoder architectures. It shows that as input and output resolutions grow, the induced kernels converge to a limiting kernel, enabling regularity assumptions independent of resolution. The authors derive upper and lower bounds for regularized stochastic gradient descent, extend the analysis to neural networks via the limiting neural tangent kernel, and provide error bounds and complexity guarantees for various kernel and encoding constructions.
By Lei Shi, Jia-Qi Yang, Ding-Xuan Zhou
arXiv:2607. 29245v1 Announce Type: cross Abstract: We study the expected improvement (EI) policy for minimizing a deterministic objective function $f$ on a nonempty compact set $\mathcal X \subset\mathbb R^d$.
By Emmanuel Vazquez, S\'ebastien Petit
arXiv:2609. 03762v1 Announce Type: new Abstract: The computation of the Bures-Wasserstein (BW) barycenter of an ensemble of positive definite matrices arises throughout machine learning, optimal transport, and quantum information.
By A. Afham
arXiv:2608. 04324v1 Announce Type: cross Abstract: This paper studies generalized low-rank matrix bandits with multiple prioritized objectives.
By Bo Xue, Ji Cheng, Haodong Jing, Hongzong Li, Shuang Qiu
arXiv:2505. 21460v2 Announce Type: replace Abstract: We study online calibration of multi-dimensional forecasts over an arbitrary convex set $P \subset \mathbb{R}^d$ relative to an arbitrary norm $|\cdot|$.
By Maxwell Fishelson, Noah Golowich, Mehryar Mohri, Jon Schneider
The paper introduces K$^2$SVD, a method that learns the leading singular functions of the Koopman operator by optimizing a Hilbert-Schmidt objective, producing a low‑rank, interpretable Koopman representation with a compact latent space. In this space, temporal evolution is modeled with a linear Gaussian state‑space model and inference is performed via Kalman filtering to reduce noise accumulation in multi‑step predictions. Experiments demonstrate that K$^2$SVD outperforms state‑of‑the‑art methods on multiple datasets, achieving faster prediction speeds and lower computational cost.
By Ruiquan Li, Yuheng Bu
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
The paper introduces a neighboring early‑stopping rule for adaptive regularization in kernel ridge regression with random features (KRR‑RF). By using a uniform grid in inverse regularization and comparing only adjacent estimators, the method reduces discrepancy checks and can be computed directly in the random‑feature space without forming the full kernel Gram matrix. Under standard source and capacity assumptions, the selected estimator achieves the oracle polynomial learning rate up to logarithmic factors, enabling regularization selection without prior knowledge of smoothness or capacity exponents.
By Caixing Wang, Zhibo Chen, Yue Wang