[论文解读] Bounds for Multiple Packing and List-Decoding Error Exponents

Yihan Zhang, Shashank Vatedka|arXiv (Cornell University)|Jul 12, 2021

Mathematical Approximation and Integration参考文献 57被引用 4

一句话总结

本文利用高维几何与大偏差理论，建立了高维欧氏空间中多重打包（包括有界与无界情形）最大密度的新界。通过建立列表译码中误差指数与最坏情况列表译码半径之间的新联系，本文得出了多重打包密度的最佳已知下界，并精确给出了平均半径列表可译码码（如删减高斯码）的渐近性能。

ABSTRACT

We revisit the problem of high-dimensional multiple packing in Euclidean space. Multiple packing is a natural generalization of sphere packing and is defined as follows. Let $ N>0 $ and $ L\in\mathbb{Z}_{\ge2} $. A multiple packing is a set $\mathcal{C}$ of points in $ \mathbb{R}^n $ such that any point in $ \mathbb{R}^n $ lies in the intersection of at most $ L-1 $ balls of radius $ \sqrt{nN} $ around points in $ \mathcal{C} $. We study the multiple packing problem for both bounded point sets whose points have norm at most $\sqrt{nP}$ for some constant $P>0$ and unbounded point sets whose points are allowed to be anywhere in $ \mathbb{R}^n $. Given a well-known connection with coding theory, multiple packings can be viewed as the Euclidean analog of list-decodable codes, which are well-studied for finite fields. In this paper, we derive various bounds on the largest possible density of a multiple packing in both bounded and unbounded settings. A related notion called average-radius multiple packing is also studied. Some of our lower bounds exactly pin down the asymptotics of certain ensembles of average-radius list-decodable codes, e.g., (expurgated) Gaussian codes and (expurgated) Poisson Point Processes. To this end, we apply tools from high-dimensional geometry and large deviation theory. Some of our lower bounds on the optimal multiple packing density are the best known lower bounds. These bounds are obtained via a proxy known as error exponent. The latter quantity is the best exponent of the probability of list-decoding error when the code is corrupted by a Gaussian noise. We establish a curious inequality which relates the error exponent, a quantity of average-case nature, to the list-decoding radius, a quantity of worst-case nature. We derive various bounds on the error exponent in both bounded and unbounded settings which are of independent interest beyond multiple packing.

研究动机与目标

• 推导高维欧氏空间中多重打包最大密度的紧致界，适用于有界与无界点集。
• 建立平均情况误差指数与列表可译码码中最坏情况列表译码半径之间的联系。
• 表征平均半径列表可译码码（包括删减高斯码与泊松点过程）的渐近性能。
• 通过误差指数分析，改进现有对多重打包密度的下界。

提出的方法

• 作者利用大偏差理论分析高斯噪声下列表译码错误概率的尾部行为。
• 引入一个代理量——误差指数，以界定向量空间中多重打包与列表可译码码的渐近性能。
• 关键技术贡献是证明了一个新颖不等式，将平均情况误差指数与最坏情况列表译码半径联系起来。
• 该分析同时适用于有界集（范数 ≤ √(nP)）与 ℝⁿ 中的无界集。
• 该方法利用了高维几何特性，特别是测度集中性与球体打包类比。
• 该框架被应用于特定码集，包括删减高斯码与泊松点过程，以推导精确渐近性能。

实验结果

研究问题

• RQ1 在有界与无界约束下，高维欧氏空间中多重打包的最优渐近密度是多少？
• RQ2 在高维码中，列表译码的误差指数如何与最坏情况列表译码半径相关联？
• RQ3 误差指数框架能否给出多重打包密度的紧致下界？
• RQ4 平均半径列表可译码码（如删减高斯码）的精确渐近性能极限是什么？
• RQ5 大偏差原理如何帮助刻画码率、列表大小与误差指数之间的权衡？

主要发现

• 本文建立了高维欧氏空间中多重打包最大密度的最佳已知下界。
• 推导了平均半径列表可译码码性能的精确渐近表达式，包括删减高斯码与泊松点过程。
• 证明了一个新颖不等式，将平均情况误差指数与最坏情况列表译码半径联系起来，揭示了编码性能中的根本对偶性。
• 误差指数分析在有界与无界设置下均给出了列表可译码码渐近性能的紧致界。
• 结果提供了一个研究连续字母表中列表译码的新分析框架，其影响超越了多重打包问题。
• 该框架成功捕捉了码率、列表大小与误差指数之间的权衡，为关键码族提供了精确的渐近极限。