Performance Analysis of Sparsity-Based Parameter Estimation

Journal article, 2017

Since the advent of the l(1) regularized least squares method (LASSO), a new line of research has emerged, which has been geared toward the application of the LASSO to parameter estimation problems. Recent years witnessed a considerable progress in this area. The notorious difficulty with discretization has been settled in the recent literature, and an entirely continuous estimation method is now available. However, an adequate analysis of this approach lacks in the current literature. This paper provides a novel analysis of the LASSO as an estimator of continuous parameters. This analysis is different from the previous ones in that our parameters of interest are associated with the support of the LASSO solution. In other words, our analysis characterizes the error in the parameterization of the support. We provide a novel framework for our analysis by studying nearly ideal sparse solutions. In this framework, we quantify the error in the high signal-to-noise ratio regime. As the result depends on the choice of the regularization parameter, our analysis also provides a new insight into the problem of selecting the regularization parameter. Without loss of generality, the results are expressed in the context of direction of arrival estimation problem.