Understanding the SPICE method and beyond

The celebrated Sparse Iterative Covariance-based Estimation (SPICE) method is analyzed in this paper by capitalizing on its equivalent reformulations as certain compressed sensing programs. Existing compressed sensing theories fall short as the considered measurement matrices in these reformulations do not satisfy the critical technical conditions such as the restricted isometry property (RIP) and the associated weights lie outside the allowable value ranges covered by the available literature. The essential observation that motivates this paper is that the reformulations take overfitting solutions under particular conditions on the measurement matrix and weights. The overfitting behaviors of these reformulations are thoroughly examined for both single measurement vector (SMV) and multiple measurement vectors (MMV) cases with identical and different noise powers. With an additional orthogonal assumption on the measurement matrix, we provide the first lower error bounds of the overfitting solutions that are shown to be tight in certain scenarios. The fundamental insights obtained in this paper not only lead to an understanding of the SPICE method but also complement the current compressed sensing research by lifting the impractical restrictions for real problem settings. The theoretical claims are demonstrated by extensive numerical experiments.