# LAP: a Linearize and Project Method for Solving Inverse Problems with Coupled Variables

## TL;DR

LAP is a flexible, Gauss-Newton-based method for efficiently solving large-scale inverse problems with coupled variables, especially when one subproblem is easier to solve, supporting various regularization and constraints.

## Contribution

The paper introduces LAP, a novel framework that linearizes and projects to solve coupled inverse problems more flexibly than existing methods like VarPro and BCD.

## Key findings

- LAP outperforms BCD and VarPro in numerical experiments.
- LAP effectively handles ill-posed imaging problems.
- LAP demonstrates flexibility in regularization and constraints.

## Abstract

Many inverse problems involve two or more sets of variables that represent different physical quantities but are tightly coupled with each other. For example, image super-resolution requires joint estimation of the image and motion parameters from noisy measurements. Exploiting this structure is key for efficiently solving these large-scale optimization problems, which are often ill-conditioned. In this paper, we present a new method called Linearize And Project (LAP) that offers a flexible framework for solving inverse problems with coupled variables. LAP is most promising for cases when the subproblem corresponding to one of the variables is considerably easier to solve than the other. LAP is based on a Gauss-Newton method, and thus after linearizing the residual, it eliminates one block of variables through projection. Due to the linearization, this block can be chosen freely. Further, LAP supports direct, iterative, and hybrid regularization as well as constraints. Therefore LAP is attractive, e.g., for ill-posed imaging problems. These traits differentiate LAP from common alternatives for this type of problem such as variable projection (VarPro) and block coordinate descent (BCD). Our numerical experiments compare the performance of LAP to BCD and VarPro using three coupled problems whose forward operators are linear with respect to one block and nonlinear for the other set of variables.