Single-loop methods for bilevel parameter learning in inverse imaging

TL;DR

This paper introduces flexible single-loop bilevel optimization methods for inverse imaging, improving computational efficiency by interweaving inner and outer steps, demonstrated on deblurring and MRI problems.

Contribution

It extends single-loop bilevel optimization to include standard inverse imaging algorithms and interweaves steps with linear solvers for enhanced performance.

Findings

Improved efficiency in hyperparameter learning for inverse imaging.

Effective interweaving of optimization and linear solver steps.

Successful application to deblurring and MRI subsampling problems.

Abstract

Bilevel optimisation is used in inverse imaging problems for hyperparameter learning/identification and experimental design, for instance, to find optimal regularisation parameters and forward operators. However, computationally, the process is costly. To reduce this cost, recently so-called single-loop approaches have been introduced. On each step of an outer optimisation method, they take just a single gradient step towards the solution of the inner problem. In this paper, we flexibilise the inner algorithm to include standard methods in inverse imaging. Moreover, as we have recently shown, significant performance improvements can be obtained in PDE-constrained optimisation by interweaving the steps of conventional iterative linear system solvers with the optimisation method. We now demonstrate how the adjoint equation in bilevel problems can also benefit from such interweaving. We evaluate the performance of our approach on identifying the deconvolution kernel for image deblurring, and the subsampling operator for magnetic resonance imaging (MRI).