基于均值的小距离轨迹重建算法的局限性
Limitations of Mean-Based Algorithms for Trace Reconstruction at Small Distance
摘要：
迹重建考虑了给定多个独立“迹”，即，通过以一定概率$p$随机且独立地删除$x$的每个符号而获得的$x$的序列。恢复长度为$n$的字符串所需的迹线数量的信息论极限仍然是未知的。该限制基本上与确定所需的跟踪数相同，给定字符串$x$和$y$以及其中之一的跟踪，哪个字符串是源。研究最多的一类算法的最坏情况下的版本的问题是“均值为基础的”算法。这是一类受限的分类器，仅使用给定样本上每个坐标的平均值。在这项工作中，我们研究了基于平均值的算法在小汉明或编辑距离字符串的限制。我们表明，一方面，区分字符串附近的汉明距离是“容易”的，这样的observers。另一方面，我们表明，区分字符串附近的编辑距离是“硬”的平均值为基础的算法。一路上，我们还描述了一个连接到著名的Prouhet-Tarry-Escott（PTE）问题，这表明了一个障碍，找到明确的难以区分的字符串：即这样的字符串将意味着明确的短期解决方案的PTE问题，一个众所周知的困难问题，在数论。此外，我们表明，反过来也是正确的，因此，寻找显式解决方案的PTE问题是等价的问题，找到明确的字符串是难以区分的平均值为基础的算法。我们的技术依赖于复分析参数，涉及仔细的三角估计，和代数技术，包括应用笛卡尔的规则的符号多项式的实数。
英文摘要：
Trace reconstruction considers the task of recovering an unknown string $x \in \{0,1\}^n$ given a number of independent "traces", i.e., subsequences of $x$ obtained by randomly and independently deleting every symbol of $x$ with some probability $p$. The information-theoretic limit of the number of traces needed to recover a string of length $n$ is still unknown. This limit is essentially the same as the number of traces needed to determine, given strings $x$ and $y$ and traces of one of them, which string is the source. The most-studied class of algorithms for the worst-case version of the problem are "mean-based" algorithms. These are a restricted class of distinguishers that only use the mean value of each coordinate on the given samples. In this work we study limitations of mean-based algorithms on strings at small Hamming or edit distance. We show that, on the one hand, distinguishing strings that are nearby in Hamming distance is "easy" for such distinguishers. On the other hand, we show that distinguishing strings that are nearby in edit distance is "hard" for mean-based algorithms. Along the way, we also describe a connection to the famous Prouhet-Tarry-Escott (PTE) problem, which shows a barrier to finding explicit hard-to-distinguish strings: namely such strings would imply explicit short solutions to the PTE problem, a well-known difficult problem in number theory. Furthermore, we show that the converse is also true, thus, finding explicit solutions to the PTE problem is equivalent to the problem of finding explicit strings that are hard-to-distinguish by mean-based algorithms. Our techniques rely on complex analysis arguments that involve careful trigonometric estimates, and algebraic techniques that include applications of Descartes' rule of signs for polynomials over the reals.
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