标题： 关于高斯噪声信道中的最小均值$p$-阶误差及其应用 显示英文标题

标题： On the Minimum Mean $p$-th Error in Gaussian Noise Channels and its Applications

摘要： 从观测值中估计任意随机向量的问题，其中噪声是加性高斯白噪声，成本函数取为最小均值$p$-阶误差（MMPE），被考虑。经典的最小均方误差（MMSE）是MMPE的一个特例。推导并讨论了MMPE的几个界、性质和应用。对于高斯和二进制输入分布，找到了最优的MMPE估计器。推导了MMPE作为输入分布、信噪比（SNR）和阶数$p$的函数的性质。特别是，证明了MMPE是$p$和SNR的连续函数。这些结果可能得益于对MMPE的插值和测度变换界。"单交叉点性质"（SCPP）在所有SNR值{\it 以上}某个已知MMSE值的情况下限制了MMSE，同时结合I-MMSE关系，是信息论中推导逆向证明的强大工具。通过研究条件MMPE的概念，展示了SCPP的统一证明（即对于任何$p$）。然后展示了SCPP的互补界，该界在所有SNR值{\it 下方}某个已知MMPE值的情况下限制了MMPE。作为MMPE的第一个应用，提供了一个基于MMPE的条件微分熵的界，这进而得到了离散输入在高斯噪声信道上实现的互信息的Ozarow-Wyner下界的推广。作为第二个应用，MMPE被证明优于之前对相变现象的描述，该现象在达到容量的码长趋于无穷大的极限时表现为MMSE作为SNR函数的不连续性。作为最终应用，MMPE被用来显示互信息的二阶导数的界，这些界比以前已知的界更紧。 显示英文摘要

摘要： The problem of estimating an arbitrary random vector from its observation corrupted by additive white Gaussian noise, where the cost function is taken to be the Minimum Mean $p$-th Error (MMPE), is considered. The classical Minimum Mean Square Error (MMSE) is a special case of the MMPE. Several bounds, properties and applications of the MMPE are derived and discussed. The optimal MMPE estimator is found for Gaussian and binary input distributions. Properties of the MMPE as a function of the input distribution, SNR and order $p$ are derived. In particular, it is shown that the MMPE is a continuous function of $p$ and SNR. These results are possible in view of interpolation and change of measure bounds on the MMPE. The `Single-Crossing-Point Property' (SCPP) that bounds the MMSE for all SNR values {\it above} a certain value, at which the MMSE is known, together with the I-MMSE relationship is a powerful tool in deriving converse proofs in information theory. By studying the notion of conditional MMPE, a unifying proof (i.e., for any $p$) of the SCPP is shown. A complementary bound to the SCPP is then shown, which bounds the MMPE for all SNR values {\it below} a certain value, at which the MMPE is known. As a first application of the MMPE, a bound on the conditional differential entropy in terms of the MMPE is provided, which then yields a generalization of the Ozarow-Wyner lower bound on the mutual information achieved by a discrete input on a Gaussian noise channel. As a second application, the MMPE is shown to improve on previous characterizations of the phase transition phenomenon that manifests, in the limit as the length of the capacity achieving code goes to infinity, as a discontinuity of the MMSE as a function of SNR. As a final application, the MMPE is used to show bounds on the second derivative of mutual information, that tighten previously known bounds.