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On Learning the Optimal Regularization Parameter in Inverse Problems

Jonathan Chirinos Rodriguez, Ernesto De Vito, Cesare Molinari, Lorenzo Rosasco, Silvia Villa

TL;DR

This paper addresses the challenge of choosing the optimal regularization parameter in inverse problems by proposing a data-driven approach grounded in empirical risk minimization. It provides a theoretical analysis showing that, with sufficient data, the method attains sharp convergence rates and adapts to unknown noise levels and problem smoothness. Numerical simulations corroborate the theory and illustrate practical performance. Overall, the work strengthens the theoretical foundation for data-driven parameter selection in inverse problems and offers a principled path toward reliable, adaptive regularization in practice.

Abstract

Selecting the best regularization parameter in inverse problems is a classical and yet challenging problem. Recently, data-driven approaches have become popular to tackle this challenge. These approaches are appealing since they do require less a priori knowledge, but their theoretical analysis is limited. In this paper, we propose and study a statistical machine learning approach, based on empirical risk minimization. Our main contribution is a theoretical analysis, showing that, provided with enough data, this approach can reach sharp rates while being essentially adaptive to the noise and smoothness of the problem. Numerical simulations corroborate and illustrate the theoretical findings. Our results are a step towards grounding theoretically data-driven approaches to inverse problems.

On Learning the Optimal Regularization Parameter in Inverse Problems

TL;DR

This paper addresses the challenge of choosing the optimal regularization parameter in inverse problems by proposing a data-driven approach grounded in empirical risk minimization. It provides a theoretical analysis showing that, with sufficient data, the method attains sharp convergence rates and adapts to unknown noise levels and problem smoothness. Numerical simulations corroborate the theory and illustrate practical performance. Overall, the work strengthens the theoretical foundation for data-driven parameter selection in inverse problems and offers a principled path toward reliable, adaptive regularization in practice.

Abstract

Selecting the best regularization parameter in inverse problems is a classical and yet challenging problem. Recently, data-driven approaches have become popular to tackle this challenge. These approaches are appealing since they do require less a priori knowledge, but their theoretical analysis is limited. In this paper, we propose and study a statistical machine learning approach, based on empirical risk minimization. Our main contribution is a theoretical analysis, showing that, provided with enough data, this approach can reach sharp rates while being essentially adaptive to the noise and smoothness of the problem. Numerical simulations corroborate and illustrate the theoretical findings. Our results are a step towards grounding theoretically data-driven approaches to inverse problems.
Paper Structure (29 sections, 2 theorems, 7 equations, 2 figures, 2 tables, 1 algorithm)

This paper contains 29 sections, 2 theorems, 7 equations, 2 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Class options
  3. Front matter
  4. Cross references and hyperlinks
  5. Cleveref
  6. Hyperef
  7. Math and equations
  8. Theorem-like environments
  9. Tables
  10. Figures
  11. Algorithms
  12. Sections
  13. Supplemental material
  14. Template
  15. Bibliography
  16. ...and 14 more sections

Key Result

Theorem 6.1

\newlabelthm:mvt0 Suppose $f$ is a function that is continuous on the closed interval $[a,b]$. and differentiable on the open interval $(a,b)$. Then there exists a number $c$ such that $a < c < b$ and In other words, $f(b)-f(a) = f'(c)(b-a)$.

Figures (2)

  • Figure 1: Example figure using external image files.
  • Figure 2: Example PGFPLOTS figure.

Theorems & Definitions (5)

  • Theorem 6.1: Mean Value Theorem
  • Corollary 6.2
  • Proof 1
  • Claim 6.3
  • Proof 2: Proof of main theorem