Bilinear Compressed Sensing - Efficiency, Structure, and Robustness

实际的数据采集过程通常依赖于未校准的系统。这是双线性压缩感知问题的自然来源。这些是测量结果线性依赖于信号和校准参数的问题。如果使用传统的压缩感知（CS）方案来处理此类双线性问题，则需要以次优的感知速率进行操作，或者由于模型失配而产生显著的重构误差。因此，在过去几年中，此类“盲信息检索问题”的理论基础的工作已经开始，部分是在我们优先计划第一阶段的项目背景下。 在许多情况下，人们已经确定，这些问题确实可以有易于处理的解决办法。 然而，这些最初的结果在实践中仍然有些不适用。 我们项目的目标是通过开发双线性压缩传感理论来缩小这一差距，以更好地解决应用中出现的问题。在这种情况下，我们确定了以下挑战，这些挑战将作为我们项目的指导主题。(1)效率：设计和分析可以科普实际问题规模的算法。 这将经常需要超越凸优化的框架。 (2)结构形式：少依赖高度随机化的结构，这些结构通常比较容易进行数学分析，但往往不切实际。(3)鲁棒性：重点关注重建算法承受现实应用中存在的噪声和模型失配影响的能力。

Practical data acquisition processes often rely on uncalibrated systems. This is a natural source of bilinear compressed sensing problems. These are problems, where the measurement outcomes depend linearly on both the signal and the calibration parameters. If one uses traditional Compressed Sensing (CS) schemes for such bilinear problems, one needs to operate at sub-optimal sensing rates or incur significant reconstruction errors due to model mismatch.For this reason, work on a theoretical foundation of such "blind information retrieval problems" has started over the past years, partly in the context of our project in the first phase of the priority program. In many cases, it has been established that these problems can indeed have tractable solutions. These first results, however, were still somewhat removed from being applicable in practice. The goal of our project will be to close this gap by developing theory for bilinear compressed sensing that better addresses issues arising in applications. In this vein, we have identified the following challenges that will serve as a guiding theme for our project. (1) Efficiency: Design and analyze algorithms that can cope with real-world problem sizes. This will frequently necessitate going beyond the framework of convex optimization. (2) Structure: Rely less on highly randomized constructions that are typically comparativelysimple to analyze mathematically, but often impractical to implement.(3) Robustness: Focus on the ability of reconstruction algorithms to withstand theimpact of noise and model mismatch present in real-world applications.