Low-rank matrix recovery problems arise naturally as mathematical formulations of various inverse problems, such as matrix completion, blind deconvolution, and phase retrieval. Over the last two decades, a number of works have rigorously analyzed the reconstruction performance for such scenarios, giving rise to a rather general understanding of the potential and the limitations of low-rank matrix models in sensing problems. In this article, we compare the two main proof techniques that have been paving the way to a rigorous analysis, discuss their potential and limitations, and survey their successful applications. On the one hand, we review approaches based on descent cone analysis, showing that they often lead to strong guarantees even in the presence of adversarial noise, but face limitations when it comes to structured observations. On the other hand, we discuss techniques using approximate dual certificates and the golfing scheme, which are often better suited to deal with practical measurement structures, but sometimes lead to weaker guarantees. Lastly, we review recent progress towards analyzing descent cones also for structured scenarios -- exploiting the idea of splitting the cones into multiple parts that are analyzed via different techniques.

中文翻译：

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鲁棒低秩矩阵恢复的证明方法

低秩矩阵恢复问题作为各种逆问题的数学公式自然出现，例如矩阵补全、盲解卷积和相位检索。在过去的二十年中，许多工作严格分析了此类场景的重建性能，从而对低秩矩阵模型在传感问题中的潜力和局限性有了相当普遍的理解。在本文中，我们比较了为严格分析铺平道路的两种主要证明技术，讨论了它们的潜力和局限性，并调查了它们的成功应用。一方面，我们回顾了基于下降锥分析的方法，表明即使在存在对抗性噪声的情况下，它们通常也能提供强有力的保证，但在结构化观察方面面临局限性。另一方面，我们讨论使用近似双重证书和高尔夫方案的技术，它们通常更适合处理实际的测量结构，但有时会导致较弱的保证。最后，我们回顾了最近在分析结构化场景下的下降锥体方面的进展——利用将锥体分成多个部分的想法，这些部分通过不同的技术进行分析。