CIF:Medium:Convex Optimization for Blind Inverse Problems

处理传感器和成像数据的基本任务之一是解决逆问题，确定产生该数据的某些基本结构的性质。这类问题通常是欠确定的，这意味着未知数的数量超过了观测值的数量，在盲目设置下，这些问题甚至更加复杂，因为在到达传感器的过程中，基本结构可能会经历一些未知的变换。本项目考虑了许多这样的盲逆问题，包括未知的点扩展函数随时间变化的非平稳反卷积；多频带信号识别，其中线谱估计被扩展到具有多个窄频带的信号；超分辨率雷达成像，其中扩展的和加速的目标可能导致延迟-多普勒空间中不想要的扩展；以及同时的盲反卷积和相位恢复。按照惯例，所有这些问题都是分开研究的。该项目在统一的优化和分析框架下共同研究所有问题，通过使用子空间对未知变换算子进行建模，将每个非凸逆问题转化为恢复结构化信号的线性逆问题。这个信号是由原始原子组产生的提升原子的简约混合物，允许研究人员使用新的原子规范来加强它的简单性。研究人员开发了新颖的分析技术来证明这一框架的最优性，方法是推导出达到信息理论极限的抽样复杂性，计算与极小极大率匹配的均方去噪误差，并开发接近Cramer-Rao界的参数估计界。该项目建立在研究人员在信号处理、连续参数化反问题的凸优化和几何建模方面的综合专业知识的基础上。

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.