Optimally Tuned Iterative Reconstruction Algorithms for Compressed Sensing
This work provides practitioners with ready-to-use, optimally tuned algorithms for sparse recovery, eliminating the need for manual parameter selection.
The authors conducted a large-scale computational experiment to optimally tune parameters for iterative thresholding and greedy algorithms for compressed sensing, achieving 'out of the box' performance that dominates prior proposals. The optimally tuned implementations maximize the phase transition, i.e., the number of nonzeros at which the algorithm can successfully recover sparse solutions.
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.