Signal models formed as linear combinations of few atoms from an over-complete dictionary or few frame vectors from a redundant frame have become central to many applications in high dimensional signal processing and data analysis. A core question is, by exploiting the intrinsic low dimensional structure of the signal, how to design the sensing process and decoder in a way that the number of measurements is essentially close to the complexity of the signal set. This chapter provides a survey of important results in answering this question, with an emphasis on a basis pursuit like convex optimization decoder that admits a wide range of random sensing matrices. The results are quite established in the case signals are sparse in an orthonormal basis, while the case with frame sparse signals is much less explored. In addition to presenting the latest results on recovery guarantee and how few random heavier-tailed measurements fulfill these recovery guarantees, this chapter also aims to provide some insights in proof techniques. We also take the opportunity of this book chapter to publish an interesting result (Theorem 3.10) about a restricted isometry like property related to a frame.

翻译：由过完备字典中的少量原子或冗余框架中的少量框架向量线性组合形成的信号模型，已成为高维信号处理与数据分析诸多应用的核心。一个关键问题是：如何通过利用信号固有的低维结构，设计感知过程与解码器，使得测量数量本质上接近信号集的复杂度。本章综述了回答该问题的重要成果，重点讨论一类类似基追踪的凸优化解码器，该类解码器允许使用广泛的随机感知矩阵。当信号在正交基下稀疏时，相关结果已较为成熟；而对于框架稀疏信号的情形，研究则远不充分。除介绍恢复保证的最新成果以及满足这些恢复保证所需的少量重尾随机测量外，本章亦旨在提供证明技术方面的若干见解。我们亦借此书章之机，发表一项关于框架类约束等距性质的趣味结果（定理3.10）。