高斯噪声中信号估计的统计物理学：相变理论与实例
Statistical Physics of Signal Estimation in Gaussian Noise: Theory and Examples of Phase Transitions
摘要：
我们考虑的问题，信号估计（去噪）从统计机械的角度来看，使用之间的关系的最小均方误差（MMSE），估计信号，和这个信号和它的噪声版本之间的互信息。本文件主要由两部分组成。在第一，我们得到几个重要的量在这个问题领域，如MMSE，微分熵，Fisher信息，自由能，和温度的广义概念之间的物理力学关系。我们还画出了与估计问题有关的某些关系与热力学和统计物理学中的平行关系之间的类比和差异。在本文的第二部分中，我们提供了几个应用实例，在那里我们演示了如何在统计物理学中习惯的某些分析工具，证明有用的MMSE的分析。在大多数这些例子中，相应的物理-力学系统由导致相变的强相互作用组成，这反过来又反映为MMSE行为中的不规则性和不连续性（类似于阈值效应）。
英文摘要：
We consider the problem of signal estimation (denoising) from a statistical mechanical perspective, using a relationship between the minimum mean square error (MMSE), of estimating a signal, and the mutual information between this signal and its noisy version. The paper consists of essentially two parts. In the first, we derive several statistical-mechanical relationships between a few important quantities in this problem area, such as the MMSE, the differential entropy, the Fisher information, the free energy, and a generalized notion of temperature. We also draw analogies and differences between certain relations pertaining to the estimation problem and the parallel relations in thermodynamics and statistical physics. In the second part of the paper, we provide several application examples, where we demonstrate how certain analysis tools that are customary in statistical physics, prove useful in the analysis of the MMSE. In most of these examples, the corresponding statistical-mechanical systems turn out to consist of strong interactions that cause phase transitions, which in turn are reflected as irregularities and discontinuities (similar to threshold effects) in the behavior of the MMSE.
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