Parameterless Optimal Approximate Message Passing

TL;DR

This paper introduces a parameter-free approximate message passing algorithm that automatically sets thresholds at each iteration, achieving optimal reconstruction error and convergence rate without prior signal knowledge or tuning.

Contribution

It proposes a novel, fully automatic thresholding method for AMP algorithms using SURE and gradient descent, with theoretical guarantees for optimality.

Findings

Attains minimum reconstruction error.

Achieves highest convergence rate.

First to provide theoretical guarantees for parameter tuning in AMP.

Abstract

Iterative thresholding algorithms are well-suited for high-dimensional problems in sparse recovery and compressive sensing. The performance of this class of algorithms depends heavily on the tuning of certain threshold parameters. In particular, both the final reconstruction error and the convergence rate of the algorithm crucially rely on how the threshold parameter is set at each step of the algorithm. In this paper, we propose a parameter-free approximate message passing (AMP) algorithm that sets the threshold parameter at each iteration in a fully automatic way without either having an information about the signal to be reconstructed or needing any tuning from the user. We show that the proposed method attains both the minimum reconstruction error and the highest convergence rate. Our method is based on applying the Stein unbiased risk estimate (SURE) along with a modified gradient descent to find the optimal threshold in each iteration. Motivated by the connections between AMP and LASSO, it could be employed to find the solution of the LASSO for the optimal regularization parameter. To the best of our knowledge, this is the first work concerning parameter tuning that obtains the fastest convergence rate with theoretical guarantees.