Many questions of fundamental interest in today's science can be formulated as inference problems: some partial, or noisy, observations are performed over a set of variables and the goal is to recover, or infer, the values of the variables based on the indirect information contained in the measurements. For such problems, the central scientific questions are: Under what conditions is the information contained in the measurements sufficient for a satisfactory inference to be possible? What are the most efficient algorithms for this task? A growing body of work has shown that often we can understand and locate these fundamental barriers by thinking of them as phase transitions in the sense of statistical physics. Moreover, it turned out that we can use the gained physical insight to develop new promising algorithms. The connection between inference and statistical physics is currently witnessing an impressive renaissance and we review here the current state-of-the-art, with a pedagogical focus on the Ising model which, formulated as an inference problem, we call the planted spin glass. In terms of applications we review two classes of problems: (i) inference of clusters on graphs and networks, with community detection as a special case and (ii) estimating a signal from its noisy linear measurements, with compressed sensing as a case of sparse estimation. Our goal is to provide a pedagogical review for researchers in physics and other fields interested in this fascinating topic.

中文翻译：

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推理的统计物理学：阈值和算法

当今科学中的许多基本问题都可以表述为推理问题：对一组变量进行一些部分或嘈杂的观察，目标是根据包含的间接信息恢复或推断变量的值在测量中。对于这些问题，核心科学问题是：在什么条件下测量中包含的信息足以使令人满意的推理成为可能？这项任务最有效的算法是什么？越来越多的工作表明，我们通常可以通过将它们视为统计物理学意义上的相变来理解和定位这些基本障碍。此外，事实证明，我们可以利用获得的物理洞察力来开发新的有前途的算法。推理和统计物理学之间的联系目前正在见证令人印象深刻的复兴，我们在这里回顾了当前最先进的技术，重点关注 Ising 模型，该模型被表述为推理问题，我们称之为种植自旋玻璃。在应用方面，我们回顾了两类问题：（i）图和网络上的集群推断，社区检测作为特例；（ii）从其嘈杂的线性测量中估计信号，压缩感知作为稀疏的情况估计。我们的目标是为物理学和其他对这一迷人主题感兴趣的领域的研究人员提供教学评论。公式化为推理问题，我们称种植自旋玻璃。在应用方面，我们回顾了两类问题：（i）图和网络上的集群推断，社区检测作为特例；（ii）从其嘈杂的线性测量中估计信号，压缩感知作为稀疏的情况估计。我们的目标是为物理学和其他对这一迷人主题感兴趣的领域的研究人员提供教学评论。公式化为推理问题，我们称种植自旋玻璃。在应用方面，我们回顾了两类问题：（i）图和网络上的集群推断，社区检测作为特例；（ii）从其嘈杂的线性测量中估计信号，压缩感知作为稀疏的情况估计。我们的目标是为物理学和其他对这一迷人主题感兴趣的领域的研究人员提供教学评论。