Invariancy of Sparse Recovery Algorithms

In this paper, a property for sparse recovery algorithms, called invariancy, is introduced. The significance of invariancy is that the performance of the algorithms with this property is less affected when the sensing (i.e., the dictionary) is ill-conditioned. This is because for this kind of algorithms, there exists implicitly an equivalent well-conditioned problem, which is being solved. Some examples of sparse recovery algorithms will also be considered and it will be shown that some of them, such as SL0, Basis Pursuit (using interior point LP solver), FOCUSS, and hard thresholding algorithms, are invariant, and some others, like Matching Pursuit and SPGL1, are not. Then, as an application example of the invariancy property, a sparse-decomposition-based method for direction of arrival estimation is reviewed, and it is shown that if an invariant algorithm is utilized for solving the corresponding sparse recovery problem, the spatial characteristics of the sensors will have essentially no effect on the final estimation, provided that the number of sensors is large enough.