# Dynamical Functional Theory for Compressed Sensing

## TL;DR

This paper develops a theoretical framework for designing generalized AMP algorithms for compressed sensing with large invariant random matrices, ensuring their fixed points align with TAP equations and their statistics match the replica ansatz.

## Contribution

It introduces a dynamical functional approach to create AMP generalizations valid for large invariant matrices, linking fixed points to TAP equations and enabling explicit analysis of the algorithm's statistics.

## Key findings

- Derived an effective stochastic process for single-component dynamics.
- Designed memory terms to make fields Gaussian, facilitating analysis.
- Confirmed asymptotic statistics align with the replica ansatz.

## Abstract

We introduce a theoretical approach for designing generalizations of the approximate message passing (AMP) algorithm for compressed sensing which are valid for large observation matrices that are drawn from an invariant random matrix ensemble. By design, the fixed points of the algorithm obey the Thouless-Anderson-Palmer (TAP) equations corresponding to the ensemble. Using a dynamical functional approach we are able to derive an effective stochastic process for the marginal statistics of a single component of the dynamics. This allows us to design memory terms in the algorithm in such a way that the resulting fields become Gaussian random variables allowing for an explicit analysis. The asymptotic statistics of these fields are consistent with the replica ansatz of the compressed sensing problem.