中国剩余定理的鲁棒性
Robustness in Chinese Remainder Theorem
摘要：
中国剩余定理在频率估计、相位展开、编码理论和分布式数据存储等方面有着广泛的应用。由于传统的CRT是非常敏感的残差由于噪声的错误，鲁棒重建整数的问题，通过错误的残差已在文献中进行了深入的研究。为了实现整数的鲁棒重构，传统的方法有两种：一种是在模中引入公约数，另一种是直接减小动态范围。本文进一步深入研究了与CRT相关的线性空间的几何性质。呼应这两种方式引入冗余，我们提出了一个伪度量分析之间的权衡误差界和动态范围的鲁棒CRT一般。此外，我们提出了第一个强大的CRT的多个数字，以解决问题的CRT为基础的欠采样频率估计在一般情况下。基于对称多项式，我们证明了在大多数情况下，该问题可以在多项式时间内有效地求解。在本文中的工作是朝着一个完整的理论解决方案的开放问题超过20年。
英文摘要：
Chinese Remainder Theorem (CRT) has been widely studied with its applications in frequency estimation, phase unwrapping, coding theory and distributed data storage. Since traditional CRT is greatly sensitive to the errors in residues due to noises, the problem of robustly reconstructing integers via the erroneous residues has been intensively studied in the literature. In order to robustly reconstruct integers, there are two kinds of traditional methods: the one is to introduce common divisors in the moduli and the other is to directly decrease the dynamic range. In this paper, we take further insight into the geometry property of the linear space associated with CRT. Echoing both ways to introduce redundancy, we propose a pseudo metric to analyze the trade-off between the error bound and the dynamic range for robust CRT in general. Furthermore, we present the first robust CRT for multiple numbers to solve the problem of the CRT-based undersampling frequency estimation in general cases. Based on symmetric polynomials, we proved that in most cases, the problem can be solved in polynomial time efficiently. The work in this paper is towards a complete theoretical solution to the open problem over 20 years.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
