Compressive parameter estimation via K-median clustering

In recent years, compressive sensing (CS) has attracted significant attention in parameter estimation tasks, including frequency estimation, time delay estimation, and localization. In order to use CS in parameter estimation, parametric dictionaries (PDs) collect observations for a sampling of the parameter space and yield sparse representations for signals of interest when the sampling is sufficiently dense. While this dense sampling can lead to high coherence in the dictionary, it is possible to leverage structured sparsity models to prevent highly coherent dictionary elements from appearing simultaneously in the signal representations, alleviating these coherence issues. However, the resulting approaches depend heavily on a careful setting of the maximum allowable coherence; furthermore, their guarantees applied on the coefficient recovery do not translate in general to the parameter estimation task. In this paper, we propose the use of the earth mover’s distance (EMD), as applied to a pair of true and estimated coefficient vectors, to measure the error of the parameter estimation. We formally analyze the connection between the aforementioned EMD and the parameter estimation error. We theoretically show that the EMD provides a better-suited metric for the performance of PD-based parameter estimation than the commonly used Euclidean distance. Additionally, we leverage the previously described relationship between K-median clustering and EMD-based sparse approximation to develop improved PD-based parameter estimation algorithms. Finally, we present numerical experiments that verify our theoretical results and show the performance improvements obtained from the proposed compressive parameter estimation algorithms.