Approximate message-passing for convex optimization with non-separable penalties
This provides a faster optimization method for problems with non-separable penalties, useful in domains like medical imaging and signal processing, though it appears incremental as an adaptation of existing message-passing algorithms.
The paper tackles optimization problems with convex objectives containing non-separable penalties like total variation by introducing an iterative scheme based on expectation-consistent approximation and vector approximate message-passing (VAMP). The method shows faster convergence than state-of-the-art approaches like FISTA and ADMM in classification and reconstruction tasks.
We introduce an iterative optimization scheme for convex objectives consisting of a linear loss and a non-separable penalty, based on the expectation-consistent approximation and the vector approximate message-passing (VAMP) algorithm. Specifically, the penalties we approach are convex on a linear transformation of the variable to be determined, a notable example being total variation (TV). We describe the connection between message-passing algorithms -- typically used for approximate inference -- and proximal methods for optimization, and show that our scheme is, as VAMP, similar in nature to the Peaceman-Rachford splitting, with the important difference that stepsizes are set adaptively. Finally, we benchmark the performance of our VAMP-like iteration in problems where TV penalties are useful, namely classification in task fMRI and reconstruction in tomography, and show faster convergence than that of state-of-the-art approaches such as FISTA and ADMM in most settings.