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

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 image and motion parameters from noisy measurements. Exploiting this structure is key for efficiently solving large-scale problems to avoid, e.g., ill-conditioned optimization problems. 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, the block can be chosen freely and can represent quadratic as well as nonlinear variables. 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 problems such as variable projection (VarPro) and block coordinate descent (BCD). Our numerical experiments compare the performance of LAP to BCD and VarPro using four coupled problems with one quadratic and one nonlinear set of variables.