arXiv Machine Learning By Simone Brugiapaglia, Nicola Rares Franco, Nicholas H. Nelsen

A short tour of operator learning theory: Convergence rates, statistical limits, and open questions

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arXiv:2603. 00819v2 Announce Type: replace-cross Abstract: This paper surveys recent developments at the intersection of operator learning, statistical learning theory, and approximation theory.

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arXiv Machine Learning
Sep 7

The Sample Complexity of Learning Lipschitz Operators with respect to Gaussian Measures

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