CIF:Medium:Convex Optimization for Blind Inverse Problems

Colorado School Of Mines, Golden CO

Investigators

Abstract

One of the fundamental tasks in processing sensor and imaging data is to solve an inverse problem, determining the nature of some fundamental structure that produced that data. Such problems are often underdetermined, meaning that the number of unknowns exceeds the number of observations, and these problems are even more complicated in the blind setting, where the fundamental structure may undergo some unknown transformation en route to the sensor. This project considers a number of such blind inverse problems, including non-stationary deconvolution, where an unknown point spread function changes over time; multi-band signal identification, where line spectrum estimation is extended to signals with multiple narrow frequency bands; super-resolution radar imaging, where extended and accelerating targets may cause unwanted spreading in the delay-Doppler space; and simultaneous blind deconvolution and phase retrieval. Conventionally, all of these problems have been studied separately. This project investigates all of the problems jointly under a unifying optimization and analysis framework. By modeling unknown transformation operators using subspaces, the investigators transform each non-convex inverse problem into a linear inverse problem of recovering a structured signal. This signal is a parsimonious mixture of lifted atoms generated by the original atomic set, allowing the investigators to enforce its simplicity using a new atomic norm. The investigators develop novel analysis techniques to demonstrate the optimality of this framework by deriving sampling complexities that achieve the information-theoretical limits, calculating mean-squared denoising errors that match the minimax rates, and developing parameter estimation bounds that approach the Cramer-Rao bounds. The project builds on the investigators' combined expertise in signal processing, convex optimization in continuously parameterized inverse problems, and geometric modeling.