| The problem of sparse signal recovery can be formulated as a problem of finding a sparse solution X ∈ Rm to an underdetermined system of equations Y=AX+Z where A ∈ Rnxm , n < m, is the measurement matrix, Z ∈ Rn is the measurement noise, and Y ∈ Rn is the noisy measurement. Let k be the number of nonzero entries in X. Then, X is sparse when k << m.;First, we study the performance limits of support recovery, i.e., the recovery of the locations of the nonzero entries of X. By connecting sparse signal recovery to communication over a multiple access channel (MAC), we show that MAC capacity region sheds light on the performance limits of support recovery. Sufficient and necessary conditions are derived to reveal the role of the model parameters, such as the dimension of the signal m, the number of measurements n, and especially the magnitudes of nonzero entries of X, in ensuring asymptotically successful support recovery. By drawing an analogy with the single-input multiple-output MAC, the information theoretic results are extended to the multiple measurement vectors (MMV) problem to shed light on the role of multiple measurements.;Next, we propose a multiple-pass algorithmic framework which exploits the variety in the dynamic range of nonzero entries. Specifically, nonzero entries with similar magnitudes are jointly detected as a group and different groups are detected sequentially. The MultiPass Lasso algorithm and its variant are studied in detail, with experimental results demonstrating the performance improvement over their one-pass counterparts, respectively, in reconstruction accuracy and computational complexity.;We also examine two novel applications of sparse signal recovery. For robust linear regression, we model the outliers in the observations, which occur infrequently, as the nonzero entries of a sparse vector, and derive methods rooted in sparse signal recovery to tackle this problem. For adaptive filtering with sparse predictors, we employ a general diversity measure for sparse signal recovery to develop a prototype least-mean-squares (LMS) type algorithm, in which the optimization is performed in the affine scaling domain to expedite the convergence. Steady-state analysis is conducted to provide insight into the performance. As two instantiations, the pALMS and pANLMS algorithms are studied in detail. Experimental results support the potentials of sparse signal recovery techniques in both applications. |
