Consistency analysis of bilevel data-driven learning in inverse problems
This provides a method for learning regularization parameters from data in inverse problems, which is incremental as it builds on existing bilevel optimization frameworks.
The paper tackles the problem of finding regularization parameters in inverse problems by using data-driven bilevel optimization, demonstrating that the approach achieves inverse accuracy independent of ambient space dimension for linear problems and showing applicability to various linear and nonlinear examples like Darcy flow and image denoising.
One fundamental problem when solving inverse problems is how to find regularization parameters. This article considers solving this problem using data-driven bilevel optimization, i.e. we consider the adaptive learning of the regularization parameter from data by means of optimization. This approach can be interpreted as solving an empirical risk minimization problem, and we analyze its performance in the large data sample size limit for general nonlinear problems. We demonstrate how to implement our framework on linear inverse problems, where we can further show the inverse accuracy does not depend on the ambient space dimension. To reduce the associated computational cost, online numerical schemes are derived using the stochastic gradient descent method. We prove convergence of these numerical schemes under suitable assumptions on the forward problem. Numerical experiments are presented illustrating the theoretical results and demonstrating the applicability and efficiency of the proposed approaches for various linear and nonlinear inverse problems, including Darcy flow, the eikonal equation, and an image denoising example.