6478 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 65, NO. 24, DECEMBER 15, 2017 Performance Analysis of Sparsity-Based Parameter Estimation Ashkan Panahi , Member, IEEE, and Mats Viberg , Fellow, IEEE Abstract—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 es- timation 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 esti- mation method is now available. However, an adequate analysis of this approach lacks in the current literature. This paper pro- vides 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. Index Terms—Superresolution theory, performance bounds, error analysis, LASSO, atomic norm regularization, atomic decomposition, continuous LASSO, off-grid estimation. I. INTRODUCTION S PARSE representations have received significant attention in the recent signal processing literature. Data acquisition, image processing and machine learning are only a few examples among the broad range of disciplines, that employ sparse rep- resentations [1]–[5]. Among other factors, the extensive study of the Basis Pursuit (BP) [6] or the Least Absolute Shrinkage and Selection Operator (LASSO) [7] has largely contributed to the development of a strong sparsity theory, which accounts for the popularity of the sparse representations. After nearly two Manuscript received December 22, 2016; revised April 20, 2017 and July 9, 2017; accepted August 30, 2017. Date of publication September 21, 2017; date of current version October 27, 2017. The associate Editor coordinating the review of this manuscript and approving it for publication was Dr. Chandra Murthy. (Corresponding author: Ashkan Panahi.) A. Panahi is with the Electrical and Computer Engineering Department, North Carolina State University, Raleigh, NC 27695 USA (e-mail: apanahi@ ncsu.edu). M. Viberg is with the Signal Processing Group, Signals and Systems Depart- ment, Chalmers University of Technology, Gothenburg 41258, Sweden (e-mail: viberg@chalmers.se). This paper has supplementary downloadable material available at http://ieeexplore.ieee.org., provided by the author. The material includes math- ematical proofs of the results. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TSP.2017.2755602 decades, sparsity-based signal processing is still an attractive subject of research in a growing field of applications. In the recent years, sparsity-based approaches have been ex- tended toward problems of parametric nature. This movement has led to a shift in the original premise of the sparsity theory [8]–[13]. Parameter estimation is one of the areas, which can benefit from parametric sparsity-based approaches. Sensor ar- ray analysis, radar and sonar detection problems are examples of such parametric problems. One of the key incentives to ap- ply the sparsity-based approaches to the parameter estimation problems is that they can improve the resolution of the estima- tor. This means that a sparsity-based estimator can potentially distinguish parameters or objects that conventional techniques fail to capture [14], [15]. Since the advent of sparse representations in the context of parameter estimation, it has been observed that the original the- ory of sparsity is incapable of characterizing the behavior of the sparsity-based estimators. Hence, a new line of research has emerged, which has been geared toward the analysis of the sparsity-based parameter estimators. Theory of superresolution is a generic term, which refers to the outcome of this line of re- search [16]. Similar to the theory of sparsity, the superresolution theory centers on the analysis of a particular technique, which is widely known as Atomic Norm DeNoising (ANDN) or Atomic Norm Regularization (ANR) [17], [18]. ANDN is a natural ex- tension of the LASSO. It resolves the inconsistency between the LASSO and the continuous parameter spaces, which is widely known as the off-grid problem. Previous studies extend the results in the original sparsity the- ory to the case of the ANDN. However, the existing superresolu- tion theory fails to provide a fair comparison between the ANDN and the conventional estimation techniques. There are multiple reasons for such a failure, a number of which are listed below: 1) Asymptotic assumptions: Both the sparsity theory and the current superresolution theory are mainly suitable for the regime where the data dimension is asymptotically large [19], [20]. Although this case is interesting, it is certainly insufficient. There are other asymptotic cases, which are of practical interest, namely the low noise variance and the large sample size cases. Although [21] discusses large sample size in the covariance matching framework, which is different to ours, these cases, especially the high SNR scenario, have not been sufficiently discussed in the current literature. 2) Error analysis: The previous studies mainly revolve around ideal recovery. There are a number of recent attempts to provide a satisfactory description of the error, some of which 1053-587X © 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications standards/publications/rights/index.html for more information.