CIF: Small: Blind Channel Estimation and Solving Bilinear Equations by Lifting and Factoring

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The focus of this research is a fundamental problem in the acquisition and transmission of data: blind channel estimation. The goal is to develop novel and widely applicable methods for making communications and imaging robust against the types of perturbations that occur when a wireless signal travels through the air between cell phone and base station, an image passes through a uncalibrated lens, or a voice reverberates off of the walls and floors of a room. One of the most challenging aspects of these problems is that these perturbations cause the observations to be non-linear functions of the unknown variables. This research program will develop a new framework for solving and analyzing the robustness of the solutions to special classes of bilinear inverse problems, and then specialize these results to particular problems in blind channel estimation.Specifically, the investigator will study a new method for blind channel estimation based on recently developed techniques for solving constrained bilinear inverse problems. This methodology, called ?lifting? in the literature, recasts the bilinear problem as a linear matrix recovery problem with a rank constraint. This recasting allows the analytical and computational techniques from the large body of work on low rank matrix recovery to be unleashed, yielding new (very effective and scalable) algorithms for solving these classical problems, and a new mathematical understanding about when they can be tractably solved. The proposed framework will also be extended to the case where there are multiple sources present, which need to be separated as the channels are estimated.