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.
By Ben Adcock, Michael Griebel, Gregor Maier
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
arXiv:2606. 17419v1 Announce Type: new Abstract: We develop approximation and generalization error estimates for multi-input neural operators, with the output error measured in Sobolev norms.
By Yahong Yang, Zecheng Zhang, Wei Zhu, Wenjing Liao, Hao Liu
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:2402.04691v5 Announce Type: replace-cross
Abstract: This study investigates the use of stochastic gradient descent (SGD) to learn operators between general Hilbert spaces. We study weak and str...
By Lei Shi, Jia-Qi Yang
arXiv:2503. 18219v2 Announce Type: replace Abstract: This work studies the sampling complexity of learning with ReLU neural networks and neural operators.
By Philipp Grohs, Samuel Lanthaler, Margaret Trautner
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-...
By Noah Amsel, Pratyush Avi, Tyler Chen, Feyza Duman Keles, Chinmay Hegde, Cameron Musco, Christopher Musco, David Persson
arXiv:2602. 02190v2 Announce Type: replace-cross Abstract: A common approach to perform PCA on probability measures is to embed them into a Hilbert space where standard functional PCA techniques apply.
By Gachon Erell, J\'er\'emie Bigot, Elsa Cazelles
arXiv:2609.36271v1 Announce Type: new
Abstract: Recent work has established that power-law spectral conditions on data enable tight convergence bounds for deterministic gradient descent, resolving th...
By Thomas Dybdahl Ahle, Yaroslav Bulatov, Christopher De Sa, Christopher R\'e
arXiv:2609. 23529v1 Announce Type: new Abstract: Neural operators have emerged as powerful surrogates for solving partial differential equations (PDEs), yet their reliability under distribution shift remains a critical barrier to deployment.
By Hang-Cheng Dong, Pengcheng Cheng
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. 15219v1 Announce Type: new Abstract: In this work, we tackle the following question: Can neural networks trained with gradient-based methods achieve the optimal computational-statistical tradeoff in learning Gaussian single-index models?
By Siyu Chen, Beining Wu, Miao Lu, Zhuoran Yang, Tianhao Wang