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

本文研究的重点是数据采集和传输中的一个基本问题：盲信道估计。 其目标是开发新颖且广泛适用的方法，用于使通信和成像鲁棒地抵抗当无线信号在手机和基站之间的空气中传播、图像通过未校准的透镜或声音在房间的墙壁和地板上回响时发生的扰动类型。 这些问题最具挑战性的方面之一是，这些扰动导致的观察是未知变量的非线性函数。 本研究计划将发展一个新的框架来解决和分析特殊类型的双线性逆问题的解决方案的鲁棒性，然后将这些结果专门用于盲信道估计中的特定问题。具体来说，研究人员将研究一种新的方法，盲信道估计的基础上，最近开发的技术解决约束双线性逆问题。这种方法，叫做？举重？在文献中，将双线性问题转化为具有秩约束的线性矩阵恢复问题。 这种重铸允许分析和计算技术从大量的工作低秩矩阵恢复被释放，产生新的（非常有效和可扩展的）算法来解决这些经典问题，以及一个新的数学理解，当他们可以被巧妙地解决。 所提出的框架还将扩展到存在多个源的情况，这些源需要在估计信道时被分离。

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.