鲁棒的Hankel矩阵补全及去离群点方法

谱稀疏信号出现在诸如雷达成像、地震数据重建、核磁共振谱法等应用中。此类信号的完整采样是耗时的或是受到技术限制的，采样到的值往往含有离群点和噪声。因此，研究用低质量的部分采样进行信号恢复是很重要的。由于谱稀疏信号对应的Hankel矩阵是低秩的，研究的问题可转化为如何鲁棒地补全低秩的Hankel矩阵。文献中非凸的方法能处理更大型的问题，并取得更好的恢复效果。先进的非凸方法SAP可解决Hankel矩阵的补全和去离群点问题。然而，SAP的理论分析是针对修改过的、每个迭代使用新采样点的版本。在本项目中，申请人首先拟用留一法对SAP原本的迭代进行分析，消除其理论与实际的隔阂；然后利用切空间投影加速SAP迭代中奇异值分解的计算，并推导相应的恢复保证；最后将提出算法的分析推广到有噪声的情况，并在跟实验室合作产生的数据上进行检验。这种更快速且理论更完善的方法能为许多应用中的部分采样和重建提供帮助。

Spectrally sparse signals arise in many applications such as radar imaging, seismic data reconstruction, and nuclear magnetic resonance spectroscopy. However, in these applications, full sampling is either time-consuming or technically prohibited. Moreover, the observed samples are often contaminated by outliers and noise. Therefore, it is important to study how to recover the signal from such degraded partial observations. Since the spectral sparsity of a signal can be exploited by the low-rank property of the corresponding Hankel matrix, the problem considered can be transferred to the robust completion of a Hankel matrix. Related works in the literature can be classified into convex or nonconvex approaches. By comparison, nonconvex approaches are more scalable to large-scale problems, and usually produce better recovery. The state-of-art nonconvex approach SAP can provably solve the simultaneous Hankel matrix completion and outlier removal problem. However, its theoretical analysis is for a modified version of SAP that uses fresh samples in each iteration. In this proposal, we first plan to analyze the original iterative process of SAP using the leave-one-out technique, bridging the gap in theory and practice. We then propose to use the tangent space projection technique to accelerate the SVD computation in each iterate of SAP, and derive the corresponding recovery guarantee. Finally, we would extend the analysis of the proposed method to the noisy setting, and validate its performances with real-world data produced from lab collaboration. Such a faster, and more theoretically sound method would be very useful for the partial sampling and robust recovery problem in many applications.