On the Relation Between Sparse Reconstruction and Parameter Estimation With Model Order Selection

We examine the relationship between sparse linear reconstruction and the classic problem of continuous parametric modeling. In sparse reconstruction, one wishes to recover a sparse amplitude vector from a measurement that is described as a linear combination of a small number of discrete additive components. Recent results in the compressive sensing literature have provided fast sparse reconstruction algorithms with guaranteed performance bounds for problems with certain structure. In this paper, we show an explicit connection between sparse reconstruction and parameter/order estimation and demonstrate how sparse reconstruction may be used to solve model order selection and parameter estimation problems. The structural assumption used in compressive sensing to guarantee reconstruction performance-the Restricted Isometry Property-is not satisfied in the general parameter estimation context. Nonetheless, we develop a method for selecting sparsity parameters such that sparse reconstruction mimics classic order selection criteria such as Akaike information criterion (AIC) and Bayesian information criterion (BIC). We compare the performance of the sparse reconstruction approach with traditional model order selection/parameter estimation techniques for a sinusoids-in-noise example. We find that the two methods have comparable performance in most cases, and that sparse linear modeling performs better than traditional model-based parameter/order estimation for closely spaced sinusoids with low signal-to-noise ratio.