Efficient learning methods for large-scale optimal inversion design

TL;DR

This paper develops efficient large-scale learning algorithms for optimal inversion design, enabling improved regularization parameter selection and regularizer learning in inverse problems using bi-level learning and Krylov methods.

Contribution

It introduces a general framework for learning optimal regularization parameters and regularizers, including norms and covariance-based matrices, with scalable algorithms for large-scale problems.

Findings

Learned regularizers outperform traditional methods.

Methods are robust to inexact forward operators.

Efficient algorithms enable large-scale application.

Abstract

In this work, we investigate various approaches that use learning from training data to solve inverse problems, following a bi-level learning approach. We consider a general framework for optimal inversion design, where training data can be used to learn optimal regularization parameters, data fidelity terms, and regularizers, thereby resulting in superior variational regularization methods. In particular, we describe methods to learn optimal $p$ and $q$ norms for ${\rm L}^p-{\rm L}^q$ regularization and methods to learn optimal parameters for regularization matrices defined by covariance kernels. We exploit efficient algorithms based on Krylov projection methods for solving the regularized problems, both at training and validation stages, making these methods well-suited for large-scale problems. Our experiments show that the learned regularization methods perform well even when there is some inexactness in the forward operator, resulting in a mixture of model and measurement error.