Low-rank matrix recovery from structured measurements has been a topic of intense study in the last decade and many important problems like matrix completion and blind deconvolution have been formulated in this framework. An important benchmark method to solve these problems is to minimize the nuclear norm, a convex proxy for the rank. A common approach to establish recovery guarantees for this convex program relies on the construction of a so-called approximate dual certificate. However, this approach provides only limited insight in various respects. Most prominently, the noise bounds exhibit seemingly suboptimal dimension factors. In this paper we take a novel, more geometric viewpoint to analyze both the matrix completion and the blind deconvolution scenario. We find that for both these applications the dimension factors in the noise bounds are not an artifact of the proof, but the problems are intrinsically badly conditioned. We show, however, that bad conditioning only arises for very small noise levels: Under mild assumptions that include many realistic noise levels we derive near-optimal error estimates for blind deconvolution under adversarial noise.

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关于盲解卷积和矩阵补全的凸几何

在过去十年中，从结构化测量中恢复低秩矩阵一直是一个深入研究的主题，并且在该框架中已经制定了许多重要问题，例如矩阵完成和盲解卷积。解决这些问题的一个重要基准方法是最小化核范数，即秩的凸代理。为该凸程序建立恢复保证的常用方法依赖于构建所谓的近似双重证书。然而，这种方法仅在各个方面提供了有限的洞察力。最突出的是，噪声边界表现出看似次优的维度因素。在本文中，我们采用一种新颖的、更几何的观点来分析矩阵完成和盲解卷积场景。我们发现，对于这两个应用，噪声边界中的维度因子不是证明的人工产物，但问题本质上是不良条件的。然而，我们表明，只有非常小的噪声水平才会出现不良条件：在包括许多现实噪声水平的温和假设下，我们为对抗性噪声下的盲去卷积得出近乎最佳的误差估计。