The support recovery problem consists of determining a sparse subset of variables that is relevant in generating a set of observations. In this paper, we study the support recovery problem in the phase retrieval model consisting of noisy phaseless measurements, which arises in a diverse range of settings such as optical detection, X-ray crystallography, electron microscopy, and coherent diffractive imaging. Our focus is on information-theoretic fundamental limits under an approximate recovery criterion, considering both discrete and Gaussian models for the sparse non-zero entries, along with Gaussian measurement matrices. In both cases, our bounds provide sharp thresholds with near-matching constant factors in several scaling regimes on the sparsity and signal-to-noise ratio. As a key step towards obtaining these results, we develop new concentration bounds for the conditional information content of log-concave random variables, which may be of independent interest.

中文翻译：

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相位检索模型中的支持恢复：信息理论基本限制

支持恢复问题包括确定与生成一组观察值相关的变量的稀疏子集。在本文中，我们研究了由噪声无相测量组成的相位检索模型中的支撑恢复问题，这些模型出现在各种环境中，例如光学检测、X 射线晶体学、电子显微镜和相干衍射成像。我们的重点是在近似恢复标准下的信息论基本限制，同时考虑稀疏非零条目的离散和高斯模型，以及高斯测量矩阵。在这两种情况下，我们的边界在稀疏性和信噪比的几种缩放方式中提供了具有接近匹配常数因子的尖锐阈值。作为获得这些结果的关键一步，