This paper considers the fundamental limit of compressed sensing for i.i.d.
signal distributions and i.i.d. Gaussian measurement matrices. Its main
contribution is a rigorous characterization of the asymptotic mutual
information (MI) and minimum mean-square error (MMSE) in this setting. Under
mild technical conditions, our results show that the limiting MI and MMSE are
equal to the values predicted by the replica method from statistical physics.
This resolves a well-known problem that has remained open for over a decade.

本文考虑了压缩感知在独立同分布信号分布和独立同分布高斯测量矩阵情况下的基本限制，并在温和技术条件下严格描述了其渐近互信息和最小均方误差。结果表明，极限互信息和最小均方误差等于统计物理学中复制方法预测的值。这解决了一个长达十年的已知问题。

高斯矩阵压缩感知中的复制对称预测是精确的

The Replica-Symmetric Prediction for Compressed Sensing with Gaussian  Matrices is Exact

Consider the minimum mean-square error (MMSE) of estimating an arbitrary
random variable from its observation contaminated by Gaussian noise. The MMSE
can be regarded as a function of the signal-to-noise ratio (SNR) as well as a
functional of the input distribution (of the random variable to be estimated).
It is shown that the MMSE is concave in the input distribution at any given
SNR. For a given input distribution, the MMSE is found to be infinitely
differentiable at all positive SNR, and in fact a real analytic function in SNR
under mild conditions. The key to these regularity results is that the
posterior distribution conditioned on the observation through Gaussian channels
always decays at least as quickly as some Gaussian density. Furthermore, simple
expressions for the first three derivatives of the MMSE with respect to the SNR
are obtained. It is also shown that, as functions of the SNR, the curves for
the MMSE of a Gaussian input and that of a non-Gaussian input cross at most
once over all SNRs. These properties lead to simple proofs of the facts that
Gaussian inputs achieve both the secrecy capacity of scalar Gaussian wiretap
channels and the capacity of scalar Gaussian broadcast channels, as well as a
simple proof of the entropy power inequality in the special case where one of
the variables is Gaussian.

研究了从受高斯噪声污染的观测中估计任意随机变量的最小均方误差，揭示出随着信噪比的增加，MMSE 是输入随机变量分布的凸函数，并且在给定任意输入分布下都是无穷可微的；同时证明了高斯输入具有保持机密性容量和广播通道容量的能力，并在某些条件下证明熵功率不等式。

高斯噪声中的估计：最小均方误差的性质

Estimation in Gaussian Noise: Properties of the Minimum Mean-Square  Error

This paper deals with arbitrarily distributed finite-power input signals
observed through an additive Gaussian noise channel. It shows a new formula
that connects the input-output mutual information and the minimum mean-square
error (MMSE) achievable by optimal estimation of the input given the output.
That is, the derivative of the mutual information (nats) with respect to the
signal-to-noise ratio (SNR) is equal to half the MMSE, regardless of the input
statistics. This relationship holds for both scalar and vector signals, as well
as for discrete-time and continuous-time noncausal MMSE estimation. This
fundamental information-theoretic result has an unexpected consequence in
continuous-time nonlinear estimation: For any input signal with finite power,
the causal filtering MMSE achieved at SNR is equal to the average value of the
noncausal smoothing MMSE achieved with a channel whose signal-to-noise ratio is
chosen uniformly distributed between 0 and SNR.

本文提出了一种新的方法，它能够将输入输出互信息与最小均方误差之间的关系联系起来，这种关系是基于信噪比的，无论输入的统计量如何，它都适用于非因果均方误差估计。