https://hal.inria.fr/inria-00369615Maleki, ArianArianMalekiDepartment of Electrical Engineering [Stanford] - Stanford UniversityDonoho, David L.David L.DonohoDepartment of Statistics [Stanford] - Stanford UniversityFreely Available, Optimally Tuned Iterative Thresholding Algorithms for Compressed SensingHAL CCSD2009[INFO.INFO-TS] Computer Science [cs]/Signal and Image Processing[SPI.SIGNAL] Engineering Sciences [physics]/Signal and Image processingRennes, IstRĂ©mi Gribonval2009-03-20 14:50:072021-02-26 09:42:032009-03-20 15:28:09enConference papersapplication/pdf1We conducted an extensive computational experiment, lasting multiple CPU-years, to optimally select parameters for important classes of algorithms for finding sparse solutions of underdetermined systems of linear equations. We make the optimally tuned implementations freely available at sparselab.stanford.edu; they can be used 'out of the box' with no user input: 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 given class of algorithms. We verify by extensive computation the robustness of our recommendations to the amplitude distribution of the nonzero coefficients as well as the matrix ensemble defining the underdetermined system. Several specific findings are established. (a) For all algorithms the worst amplitude distribution for nonzeros is generally the constantamplitude random-sign distribution; where all nonzeros are the same size. (b) Various random matrix ensembles give the same phase transitions; random partial isometries give different transitions and require different tuning; (c) Optimally tuned Subspace Pursuit dominates optimally tuned CoSaMP, particularly so when the system is almost square. (d) For randomly decimated partial Fourier transform sampling, our recommended Iterative Soft Thresholding gives extremely good performance, making more complex algorithms like CoSaMP and Subspace Pursuit relatively pointless.