Sparse support recovery using correlation information in the presence of additive noise

The correlation based framework has recently been proposed for sparse support recovery in noiseless case. To solve this framework, the constrained least absolute shrinkage and selection operator (LASSO) was employed. The regularization parameter in the constrained LASSO was found to be a key to the recovery. This paper will discuss the sparse support recoverability via the framework and adjustment of the regularization parameter in noisy case. The main contribution is to provide noise-related conditions to guarantee the sparse support recovery. It is pointed out that the candidates of the regularization parameter taken from the noise-related region can achieve the optimization and the effect of the noise cannot be ignored. When the number of the samples is finite, the sparse support recoverability is further discussed by estimating the recovery probability for the fixed regularization parameter in the region. The asymptotic consistency is obtained in probabilistic sense when the number of the samples tends to infinity. Simulations are given to demonstrate the validity of our results.