Bilinear Compressed Sensing - Efficiency, Structure, and Robustness

实际的数据采集过程通常依赖于未经校准的系统。这是双线性压缩传感问题的自然来源。这些都是问题，其中测量结果与信号和校准参数都线性相关。如果使用传统的压缩感知(CS)方案来处理这类双线性问题，则需要以次优的感知速率运行，或者由于模型失配而导致显著的重建误差。因此，在过去的几年中，我们在优先计划的第一阶段的项目的背景下，开始了对这类“盲信息检索问题”的理论基础的研究。在许多情况下，已经确定这些问题确实可以有易于处理的解决方案。然而，这些最初的结果仍然在一定程度上不适用于实践。我们项目的目标将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。本着这一精神，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：设计和分析能够处理真实问题大小的算法。这常常需要超越凸优化的框架。(2)结构：较少依赖高度随机化的结构，这些结构在数学上通常相对简单，但在实现上往往不切实际。(3)健壮性：重点关注重建算法抵御现实世界应用中存在的噪声和模型失配的影响的能力。

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.