We study the spectral recovery problem for dynamical sampling on a finite cyclic grid. Given time snapshots obtained from a fixed uniform spatial subsampling of the orbit $x_{\ell}=A^{\ell}f$, we aim to recover the spectrum of the unknown circular convolution operator $A$. However, in the presence of outliers, even in only a few snapshots, existing approaches often struggle to recover the spectrum. We address this challenge by proposing a novel robust spectral recovery model in the presence of time-sparse corruptions. We propose a robust pipeline that lifts the problem to a sequence of robust low-rank Hankel recovery and completion tasks, followed by Prony-type spectral estimation. Numerical experiments confirm the accurate spectral recovery of the proposed approach and exhibit its superior robustness against state-of-the-art under various settings.

翻译：我们研究有限循环网格上动态采样的谱恢复问题。给定由固定均匀空间子采样得到的轨道 $x_{\ell}=A^{\ell}f$ 的时间快照，旨在恢复未知循环卷积算子 $A$ 的谱。然而，在存在异常值的情况下，即使仅有少量快照，现有方法通常难以恢复谱。针对这一挑战，我们提出了一种在时间稀疏污染情形下的新型鲁棒谱恢复模型。该模型构建了鲁棒处理流程，将问题转化为一系列鲁棒低秩汉克尔恢复与补全任务，随后采用Prony型谱估计方法。数值实验证实了所提方法在谱恢复上的准确性，并展示了其在多种设置下相较于现有最优方法的卓越鲁棒性。