[论文解读] Information-Theoretic Capacity and Error Exponents of Stationary Point Processes under Random Additive Displacements

Venkat Anantharam, François Baccelli|arXiv (Cornell University)|Dec 22, 2010

Point processes and geometric inequalities参考文献 12被引用 4

一句话总结

本文在高维欧氏空间中，针对平稳点过程在随机加性位移下的容量和错误指数，建立了精确的信息论阈值。当强度的对数比例因子与噪声微分熵率之和为负时，可通过最大似然法实现可靠解码（错误概率 → 0）；否则解码失败。通过噪声熵谱的大偏离理论推导出错误指数，得到泊松过程与马特恩过程的闭式表达式，并与香农加性噪声信道建立联系，且在高斯情形下恢复了已知界。

ABSTRACT

This paper studies the Shannon regime for the random displacement of stationary point processes. Let each point of some initial stationary point process in $\R^n$ give rise to one daughter point, the location of which is obtained by adding a random vector to the coordinates of the mother point, with all displacement vectors independently and identically distributed for all points. The decoding problem is then the following one: the whole mother point process is known as well as the coordinates of some daughter point; the displacements are only known through their law; can one find the mother of this daughter point? The Shannon regime is that where the dimension $n$ tends to infinity and where the logarithm of the intensity of the point process is proportional to $n$. We show that this problem exhibits a sharp threshold: if the sum of the proportionality factor and of the differential entropy rate of the noise is positive, then the probability of finding the right mother point tends to 0 with $n$ for all point processes and decoding strategies. If this sum is negative, there exist mother point processes, for instance Poisson, and decoding strategies, for instance maximum likelihood, for which the probability of finding the right mother tends to 1 with $n$. We then use large deviations theory to show that in the latter case, if the entropy spectrum of the noise satisfies a large deviation principle, then the error probability goes exponentially fast to 0 with an exponent that is given in closed form in terms of the rate function of the noise entropy spectrum. This is done for two classes of mother point processes: Poisson and Matérn. The practical interest to information theory comes from the explicit connection that we also establish between this problem and the estimation of error exponents in Shannon's additive noise channel with power constraints on the codewords.

研究动机与目标

• 分析高维欧氏空间中平稳点过程经随机位移后可靠通信的根本极限。
• 基于强度对数因子与噪声微分熵率之和，建立解码可靠性的精确阈值。
• 利用噪声熵谱的大偏离理论，推导出闭式错误指数。
• 通过重新参数化，将点过程模型与香农加性噪声信道（带功率约束）联系起来。
• 在高信噪比区域，恢复并推广已知的错误指数界，包括香农与加拉格尔的随机与删余错误指数。

提出的方法

• 在 R^n 中建模一个平稳点过程，其中每个点被独立同分布的噪声向量随机位移。
• 将解码问题定义为：在已知母过程与噪声分布的前提下，识别某个子点的母点。
• 应用 Palm 微积分与质量传输原理，将错误概率表示为点过程统计量的函数。
• 利用噪声熵谱的大偏离理论，推导错误概率的指数衰减率。
• 应用 G"artner--Ellis 定理与 Laplace--Varadhan 积分引理，以闭式形式计算错误指数。
• 对泊松过程与马特恩硬核点过程验证结果，显示在高斯情形下收敛至已知界。

实验结果

研究问题

• RQ1 在高维空间中，被位移点过程的可靠解码的根本阈值是什么？
• RQ2 强度的对数因子与噪声的微分熵率之和如何决定解码性能？
• RQ3 大偏离理论能否用于推导一般平稳遍历噪声过程的显式错误指数？
• RQ4 该点过程模型与带功率约束的香农加性噪声信道之间有何联系？
• RQ5 所推导的错误指数是否能恢复或推广已知结果，如香农的随机与删余错误指数？

主要发现

• 存在一个精确阈值：若强度的对数比例因子与噪声微分熵率之和为负，则错误概率随维度指数快速衰减至零；否则，错误概率趋于一。
• 对于泊松与马特恩点过程，当满足阈值条件时，最大似然解码可使错误概率趋于零。
• 在噪声熵谱满足大偏离原理的假设下，通过噪声熵谱的速率函数，以闭式形式推导出错误指数。
• 该点过程模型的错误指数可通过简单重新参数化，直接映射到香农加性噪声信道的错误指数。
• 在白高斯噪声情形下，所推导的指数在 Poltyrev 区域内同时恢复了香农的随机与删余错误指数。
• 结果为所有满足大偏离条件的平稳遍历噪声过程提供了新的、紧致的错误指数界，在高信噪比高斯情形下与最佳已知界一致。