Improving A*OMP: Theoretical and Empirical Analyses With a Novel Dynamic Cost Model

Best-first search has been recently utilized for compressed sensing (CS) by the A* orthogonal matching pursuit (A*OMP) algorithm. In this work, we concentrate on theoretical and empirical analyses of A*OMP. We present a restricted isometry property (RIP) based general condition for exact recovery of sparse signals via A*OMP. In addition, we develop online guarantees which promise improved recovery performance with the residue-based termination instead of the sparsity-based one. We demonstrate the recovery capabilities of A*OMP with extensive recovery simulations using the adaptive-multiplicative (AMul) cost model, which effectively compensates for the path length differences in the search tree. The presented results, involving phase transitions for different nonzero element distributions as well as recovery rates and average error, reveal not only the superior recovery accuracy of A*OMP, but also the improvements with the residue-based termination and the AMul cost model. Comparison of the run times indicate the speed up by the AMul cost model. We also demonstrate a hybrid of OMP and A?OMP to accelerate the search further. Finally, we run A*OMP on a sparse image to illustrate its recovery performance for more realistic coefcient distributions

Ultra-high Dimensional Multiple Output Learning With Simultaneous Orthogonal Matching Pursuit: A Sure Screening Approach

We propose a novel application of the Simultaneous Orthogonal Matching Pursuit (S-OMP) procedure for sparsistant variable selection in ultra-high dimensional multi-task regression problems. Screening of variables, as introduced in \cite{fan08sis}, is an efficient and highly scalable way to remove many irrelevant variables from the set of all variables, while retaining all the relevant variables. S-OMP can be applied to problems with hundreds of thousands of variables and once the number of variables is reduced to a manageable size, a more computationally demanding procedure can be used to identify the relevant variables for each of the regression outputs. To our knowledge, this is the first attempt to utilize relatedness of multiple outputs to perform fast screening of relevant variables. As our main theoretical contribution, we prove that, asymptotically, S-OMP is guaranteed to reduce an ultra-high number of variables to below the sample size without losing true relevant variables. We also provide formal evidence that a modified Bayesian information criterion (BIC) can be used to efficiently determine the number of iterations in S-OMP. We further provide empirical evidence on the benefit of variable selection using multiple regression outputs jointly, as opposed to performing variable selection for each output separately. The finite sample performance of S-OMP is demonstrated on extensive simulation studies, and on a genetic association mapping problem. $Keywords$ Adaptive Lasso; Greedy forward regression; Orthogonal matching pursuit; Multi-output regression; Multi-task learning; Simultaneous orthogonal matching pursuit; Sure screening; Variable selectio

Self-Dictionary Sparse Regression for Hyperspectral Unmixing: Greedy Pursuit and Pure Pixel Search are Related

This paper considers a recently emerged hyperspectral unmixing formulation based on sparse regression of a self-dictionary multiple measurement vector (SD-MMV) model, wherein the measured hyperspectral pixels are used as the dictionary. Operating under the pure pixel assumption, this SD-MMV formalism is special in that it allows simultaneous identification of the endmember spectral signatures and the number of endmembers. Previous SD-MMV studies mainly focus on convex relaxations. In this study, we explore the alternative of greedy pursuit, which generally provides efficient and simple algorithms. In particular, we design a greedy SD-MMV algorithm using simultaneous orthogonal matching pursuit. Intriguingly, the proposed greedy algorithm is shown to be closely related to some existing pure pixel search algorithms, especially, the successive projection algorithm (SPA). Thus, a link between SD-MMV and pure pixel search is revealed. We then perform exact recovery analyses, and prove that the proposed greedy algorithm is robust to noise---including its identification of the (unknown) number of endmembers---under a sufficiently low noise level. The identification performance of the proposed greedy algorithm is demonstrated through both synthetic and real-data experiments