Graphical Models Concepts in Compressed Sensing

TL;DR

This paper reviews how graphical models and message passing algorithms can be applied to large-scale compressed sensing problems, particularly LASSO, and discusses deriving fast algorithms with provable high-dimensional performance.

Contribution

It introduces the application of graphical models and message passing techniques to compressed sensing, providing new algorithms and theoretical analysis for LASSO reconstruction.

Findings

Development of fast approximate message passing algorithms

Proof of exact high-dimensional limit results for LASSO risk

Enhanced understanding of graphical models in compressed sensing

Abstract

This paper surveys recent work in applying ideas from graphical models and message passing algorithms to solve large scale regularized regression problems. In particular, the focus is on compressed sensing reconstruction via ell_1 penalized least-squares (known as LASSO or BPDN). We discuss how to derive fast approximate message passing algorithms to solve this problem. Surprisingly, the analysis of such algorithms allows to prove exact high-dimensional limit results for the LASSO risk. This paper will appear as a chapter in a book on `Compressed Sensing' edited by Yonina Eldar and Gitta Kutyniok.