Correlation Awareness in Low-Rank Models: Sampling, Algorithms, and Fundamental Limits

The role of correlation awareness in low-rank compressive inverse problems is studied in this article. In such inverse problems, the ultimate goal is to estimate certain physically meaningful parameters from measurements collected across space and time. The spatiotemporal correlation structure of the data can be judiciously exploited to design highly efficient samplers that allow reliable parameter estimation from compressed measurements. For a large class of spectrum estimation problems (including source localization and line spectrum estimation), certain structured samplers, based on the idea of difference sets, will be shown to be optimal and outperform random samplers. Using these samplers, it is even possible to localize more sources than the number of physical sensors. The underlying principles are also extended to sparse estimation problems under the Bayesian framework. Fundamental performance limits in terms of Cramer-Rao bounds (CRBs) are studied in a new underdetermined regime, and certain saturation effects that only occur in this regime are discussed.