This chapter provides a principled introduction to uncertainty relations underlying sparse signal recovery. We start with the seminal work by Donoho and Stark, 1989, which defines uncertainty relations as upper bounds on the operator norm of the band-limitation operator followed by the time-limitation operator, generalize this theory to arbitrary pairs of operators, and then develop -- out of this generalization -- the coherence-based uncertainty relations due to Elad and Bruckstein, 2002, as well as uncertainty relations in terms of concentration of $1$-norm or $2$-norm. The theory is completed with the recently discovered set-theoretic uncertainty relations which lead to best possible recovery thresholds in terms of a general measure of parsimony, namely Minkowski dimension. We also elaborate on the remarkable connection between uncertainty relations and the "large sieve", a family of inequalities developed in analytic number theory. It is finally shown how uncertainty relations allow to establish fundamental limits of practical signal recovery problems such as inpainting, declipping, super-resolution, and denoising of signals corrupted by impulse noise or narrowband interference. Detailed proofs are provided throughout the chapter.

中文翻译：

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不确定关系和稀疏信号恢复

本章对稀疏信号恢复的不确定性关系进行了原则性的介绍。我们从 Donoho 和 Stark 在 1989 年的开创性工作开始，该工作将不确定性关系定义为带限算子的算子范数的上限，然后是时间限制算子，将该理论推广到任意的算子对，然后发展——出于这种概括——由于 Elad 和 Bruckstein，2002 年基于相干性的不确定关系，以及在 $1$-norm 或 $2$-norm 集中的不确定关系。该理论是通过最近发现的集合论不确定性关系完成的，这些关系导致根据一般简约度量（即 Minkowski 维数）的最佳可能恢复阈值。我们还详细阐述了不确定关系与“大筛子”之间的显着联系，“大筛子”是解析数论中发展起来的一系列不等式。最后展示了不确定性关系如何允许建立实际信号恢复问题的基本限制，例如修复、去削波、超分辨率和被脉冲噪声或窄带干扰破坏的信号去噪。整章都提供了详细的证明。