CIF:Medium:Convex Optimization for Blind Inverse Problems

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One of the fundamental tasks in processing sensor and imaging datais to solve an inverse problem, determining the nature of somefundamental structure that produced that data. Such problems areoften underdetermined, meaning that the number of unknowns exceedsthe number of observations, and these problems are even morecomplicated in the blind setting, where the fundamental structuremay 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 pointspread function changes over time; multi-band signalidentification, where line spectrum estimation is extended tosignals with multiple narrow frequency bands; super-resolutionradar imaging, where extended and accelerating targets may causeunwanted spreading in the delay-Doppler space; and simultaneousblind deconvolution and phase retrieval. Conventionally, all ofthese problems have been studied separately. This projectinvestigates all of the problems jointly under a unifyingoptimization and analysis framework.By modeling unknown transformation operators using subspaces, theinvestigators transform each non-convex inverse problem into alinear inverse problem of recovering a structured signal. Thissignal is a parsimonious mixture of lifted atoms generated by theoriginal atomic set, allowing the investigators to enforce itssimplicity using a new atomic norm. The investigators develop novelanalysis techniques to demonstrate the optimality of this frameworkby deriving sampling complexities that achieve theinformation-theoretical limits, calculating mean-squared denoisingerrors that match the minimax rates, and developing parameterestimation bounds that approach the Cramer-Rao bounds. The projectbuilds on the investigators' combined expertise in signalprocessing, convex optimization in continuously parameterizedinverse problems, and geometric modeling.