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 using the ubiquitous Restricted Isometry Property (RIP) [9] to have optimal-order uniform recovery guarantees. However, it is unclear when the RIP-based sufficient conditions on the algorithm are satisfied. We present a framework in which this task can be achieved; translating these conditions for Gaussian measurement matrices into requirements on the signal’s sparsity level, size and number of measurements. We illustrate this approach on three of the state-of-the-art greedy algorithms: CoSaMP [27], Subspace Pursuit (SP) [11] and Iterated Hard Thresholding (IHT) [6]. Designed to allow a direct comparison of existing theory, our framework implies that IHT, the lowest of the three in computational cost, also requires fewer compressed sensing measurements than CoSaMP and SP.