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

Simone Brugiapaglia, Nicola Rares Franco, Nicholas H. Nelsen

TL;DR

This paper surveys recent developments at the intersection of operator learning, statistical learning theory, and approximation theory with a focus on holomorphic operators and neural network approximations and illustrates fundamental performance limits in terms of sample size.

Abstract

This paper surveys recent developments at the intersection of operator learning, statistical learning theory, and approximation theory. First, it reviews error bounds for empirical risk minimization with a focus on holomorphic operators and neural network approximations. Next, it illustrates fundamental performance limits in terms of sample size by adopting a minimax perspective and considering various notions of regularity beyond holomorphy. The paper ends with a discussion on the interplay between these two perspectives and related open questions.

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

TL;DR

This paper surveys recent developments at the intersection of operator learning, statistical learning theory, and approximation theory with a focus on holomorphic operators and neural network approximations and illustrates fundamental performance limits in terms of sample size.

Abstract

This paper surveys recent developments at the intersection of operator learning, statistical learning theory, and approximation theory. First, it reviews error bounds for empirical risk minimization with a focus on holomorphic operators and neural network approximations. Next, it illustrates fundamental performance limits in terms of sample size by adopting a minimax perspective and considering various notions of regularity beyond holomorphy. The paper ends with a discussion on the interplay between these two perspectives and related open questions.
Paper Structure (4 sections, 6 theorems, 22 equations)

This paper contains 4 sections, 6 theorems, 22 equations.

Table of Contents

  1. Introduction
  2. Error bounds for empirical risk minimization
  3. Minimax analysis of operator learning
  4. Conclusions and open problems

Key Result

theorem 1

Instate Assumption assumption:setting. Assume that for some $r>1$ and $R>0$, the operator $\mathscr{G}$ admits an holomorphic extension $\tilde{\mathscr{G}}$ over a complex open set $\mathcal{O}$ containing $\mathscr{E}^{-1}_{\infty}(\mathsf{H}_{r,R})$. Further assume that for some $t>0$, Let $\kappa\coloneqq2\min\{r-1,t\}$ and fix $\tau>0$ arbitrarily small. Then there exist positive constants $

Theorems & Definitions (6)

  • theorem 1
  • theorem 2
  • theorem 3
  • theorem 4
  • theorem 5
  • theorem 6