Optimally Tuned Iterative Reconstruction Algorithms for Compressed Sensing

TL;DR

This paper presents extensively optimized iterative algorithms for compressed sensing that automatically achieve the best possible phase transition performance without user tuning, outperforming existing methods.

Contribution

It introduces a comprehensive computational tuning process for various sparse recovery algorithms, enabling out-of-the-box optimal performance based on phase transition analysis.

Findings

Optimally tuned algorithms outperform existing methods.

Phase transition is a well-defined measure for algorithm performance.

The tuning achieves the highest possible success thresholds.

Abstract

We conducted an extensive computational experiment, lasting multiple CPU-years, to optimally select parameters for two important classes of algorithms for finding sparse solutions of underdetermined systems of linear equations. We make the optimally tuned implementations available at {\tt sparselab.stanford.edu}; they run `out of the box' with no user tuning: it is not necessary to select thresholds or know the likely degree of sparsity. Our class of algorithms includes iterative hard and soft thresholding with or without relaxation, as well as CoSaMP, subspace pursuit and some natural extensions. As a result, our optimally tuned algorithms dominate such proposals. Our notion of optimality is defined in terms of phase transitions, i.e. we maximize the number of nonzeros at which the algorithm can successfully operate. We show that the phase transition is a well-defined quantity with our suite of random underdetermined linear systems. Our tuning gives the highest transition possible within each class of algorithms.