The paper investigates how many linear samples are needed to learn Lipschitz operators under Gaussian measures. It establishes both lower and upper bounds on the Hermite polynomial approximation error and shows that the minimal worst‑case error cannot converge algebraically with the number of samples. However, if the covariance operator of the Gaussian measure decays rapidly, convergence rates arbitrarily close to any algebraic rate can be achieved.
arXiv:2607. 11921v1 Announce Type: cross Abstract: A key question in operator learning is how to design surrogate operators with provable approximation guarantees in reasonable computational time.
arXiv:2606. 17319v1 Announce Type: cross Abstract: Motivated by the optimization of bounded binary black-box functions, we study the problem of learning polynomial surrogates over the Boolean hypercube.
arXiv:2503. 18219v2 Announce Type: replace Abstract: This work studies the sampling complexity of learning with ReLU neural networks and neural operators.
arXiv:2606. 27298v1 Announce Type: cross Abstract: We study the fundamental problem of learning a high-dimensional Gaussian truncated to an unknown halfspace.
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...
arXiv:2511. 11498v2 Announce Type: replace-cross Abstract: We consider the problems of \emph{learning} and \emph{testing} real-valued convex functions over Gaussian space.
arXiv:2607. 07468v1 Announce Type: cross Abstract: We study the recovery of sparse functions from finite, noisy, and indirect observations in the framework of statistical inverse learning.
arXiv:2507.19290v2 Announce Type: replace-cross Abstract: We study the problem of learning a structured approximation (low-rank, sparse, banded, etc.) to an unknown matrix $A$ given access to matrix-...
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.
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.
arXiv:2606. 01244v2 Announce Type: replace-cross Abstract: Inspired by the function-space theory of neural networks, we formulate and analyze a variation space for nonlinear operators between Hilbert spaces, defined through vector-valued Borel measures of bounded variation.
arXiv:2610.18166v1 Announce Type: cross Abstract: Randomly initialised neural networks of width $m$ converge to Gaussian processes as $m \to \infty$, with two types of quantitative bounds characteris...