Phase Transitions for Greedy Sparse Approximation Algorithms

A major enterprise in compressed sensing and sparse approximation is the design and analysis of computationally tractable algorithms for recovering sparse, exact or approximate, solutions of underdetermined linear systems of equations. Many such algorithms have now been proven to have optimal-order uniform recovery guarantees using the ubiquitous Restricted Isometry Property (RIP) [11]. However, without specifying a matrix, or class of matrices, it is unclear when the RIP-based sufficient conditions on the algorithm are satisfied. Bounds on RIP constants can be inserted into the algorithms RIP based conditions, translating the conditions into requirements on the signal's sparsity level, length, and number of measurements. We illustrate this approach for Gaussian matrices on three of the state-of-the-art greedy algorithms: CoSaMP [29], Subspace Pursuit (SP) [13] and Iterative Hard Thresholding (IHT) [8]. Designed to allow a direct comparison of existing theory, our framework impli...

• Type: Journal
• Year: 2010
• Where: CORR
• Authors: Jeffrey D. Blanchard, Coralia Cartis, Jared Tanner, Andrew Thompson
• Keywords: Algorithms, Compressed Sensing, CORR 2010, Education, Isometry Property