Principles of Modern Radar - Volume 2 1891121537

5.3 SR Algorithms 167than the norm of the true signal, suggesting complete failure. See [37] for an excellentdiscussion of this issue. This example perhaps suggests why one might consider using agiven SR algorithm even when conditions like RIP are not met, provided that the signalof interest is still sparse or nearly so. Performance guarantees for these sample parameterproblems remain at least a partially open research problem, although some progress hasbeen made for the specific DFT case in [59].We shall consider several classes of SR algorithms along with examples. New algorithmsare being developed and extended at a breathtaking pace in the CS literature. Indeed,it is not at all uncommon to see articles improving and extending other articles that arestill available only as preprints. In some cases, multiple generations of this phenomenonare observed. Thus, while several current state-of-the-art algorithms will be referenced,online references can be consulted for new developments before actually employing thesetechniques. The good news is that a wide range of excellent SR algorithms for both generaland fairly specific applications are available online for download and immediate use. 21Furthermore, part of the beauty of these SR algorithms is their general simplicity. Severalhighly accurate algorithms can be coded in just a few dozen lines in a language likeMATLAB.Before delving into the collection of algorithms, we will make a few comments aboutthe required inputs. The SR algorithms we survey will typically require a handful ofparameters along with the data y. Most of the algorithms will require either an explicitestimate of s or a regularization parameter that is implicitly related to the sparsity. Inmany cases, these parameters can be selected with relative ease. In addition, many ofthe algorithms are amenable to warm-starting procedures, where a parameter is variedslowly. The repeated solutions are greatly accelerated by using the solution for a similarparameter setting to initialize the algorithm. One of the many algorithms leveraging thisidea is Nesterov’s Algorithm (NESTA) [61].Finally, an issue of particular importance is the required information about the forwardoperator A. Obviously, providing the matrix A itself allows any required computations tobe completed. However, in many cases, A represents a transform like the DFT or discretewavelet transform (DWT), or some other operator that can be implemented without explicitcalculation and storage of the matrix A. Since the A matrix can easily be tens or evenhundreds of gigabytes in some interesting problems, the ability to perform multiplicationswith A and A H without explicit storage is essential. While some of the SR algorithmsrequire explicit access to A itself, many first-order algorithms require only the ability tomultiply a given vector with A and A H . We will focus primarily on these algorithms, sincethey are the only realistic approaches for many large-scale radar problems.We will divide our discussion of SR algorithms into a series of subsections. First, wewill discuss penalized least squares methods for solving variants of (5.12). We will then turnto fast iterative thresholding methods and closely related reweighting techniques. All ofthese approaches have close ties to (5.12). In contrast, greedy methods leverage heuristicsto obtain very fast algorithms with somewhat weaker performance guarantees. Finally,our discussion will briefly address Bayesian approaches to CS, methods for incorporatingsignal structure beyond simple sparsity, and approaches for handling uncertainty in theforward operator A.21 An excellent list is maintained in [60].