Title: High SNR Consistent Compressive Sensing,
Abstract: High signal to noise ratio (SNR) consistency of model selection criteria in
linear regression models has attracted a lot of attention recently. However,
most of the existing literature on high SNR consistency deals with model order
selection. Further, the limited literature available on the high SNR
consistency of subset selection procedures (SSPs) is applicable to linear
regression with full rank measurement matrices only. Hence, the performance of
SSPs used in underdetermined linear models (a.k.a compressive sensing (CS)
algorithms) at high SNR is largely unknown. This paper fills this gap by
deriving necessary and sufficient conditions for the high SNR consistency of
popular CS algorithms like $l_0$-minimization, basis pursuit de-noising or
LASSO, orthogonal matching pursuit and Dantzig selector. Necessary conditions
analytically establish the high SNR inconsistency of CS algorithms when used
with the tuning parameters discussed in literature. Novel tuning parameters
with SNR adaptations are developed using the sufficient conditions and the
choice of SNR adaptations are discussed analytically using convergence rate
analysis. CS algorithms with the proposed tuning parameters are numerically
shown to be high SNR consistent and outperform existing tuning parameters in
the moderate to high SNR regime.