arXiv 2004.068282020-04cs.DS

一种改进的删除通道种群恢复算法

Improved Algorithms for Population Recovery from the Deletion Channel

Shyam Narayanan

备注：: 30 pages. To appear in Symposium on Discrete Algorithms (SODA), 2021

摘要：

人口恢复问题要求人们恢复一个未知的分布在$n$位字符串访问独立的噪声样本的字符串从分布。最近，Ban等人[BCF+19]研究了通过删除通道诱导噪声的问题。这个问题概括了著名的迹重建问题，其中一个希望在删除通道下学习单个字符串。Ban等人展示了如何使用$\exp\big(n^{1/2} \cdot (\log n)^{O(\ell)}\big)$样本学习字符串上的$\ell$稀疏分布。在这项工作中，我们只使用$\exp\big(\tilde{O}(n^{1/3}) \cdot \ell^2\big)$样本学习分布，通过开发[DOS 17，NP 17]的算法的更高阶模拟，该算法解决了$\exp\big(\tilde{O}(n^{1/3})\big)$样本中的轨迹重建。我们还给出了在$n$中的第一个具有运行时间次指数的算法，解决了$\exp\big(\tilde{O}(n^{1/3}) \cdot \ell^3\big)$样本和时间上的种群恢复问题。值得注意的是，当$\ell = O(1)$时，我们对$n$的依赖性几乎匹配[DOS 17，NP 17]的上界，并且我们将对$\ell$的依赖性从双指数降低到单指数。因此，我们能够学习大量的字符串混合：而Ban et al.的算法只能学习具有次指数数量样本的$O(\log n/\log \log n)$字符串的混合，我们能够学习$\exp\big(n^{1/3 + o(1)}\big)$样本和时间中的$n^{o(1)}$字符串的混合。

英文摘要：

The population recovery problem asks one to recover an unknown distribution over $n$-bit strings given access to independent noisy samples of strings drawn from the distribution. Recently, Ban et al. [BCF+19] studied the problem where the noise is induced through the deletion channel. This problem generalizes the famous trace reconstruction problem, where one wishes to learn a single string under the deletion channel. Ban et al. showed how to learn $\ell$-sparse distributions over strings using $\exp\big(n^{1/2} \cdot (\log n)^{O(\ell)}\big)$ samples. In this work, we learn the distribution using only $\exp\big(\tilde{O}(n^{1/3}) \cdot \ell^2\big)$ samples, by developing a higher-moment analog of the algorithms of [DOS17, NP17], which solve trace reconstruction in $\exp\big(\tilde{O}(n^{1/3})\big)$ samples. We also give the first algorithm with a runtime subexponential in $n$, solving population recovery in $\exp\big(\tilde{O}(n^{1/3}) \cdot \ell^3\big)$ samples and time. Notably, our dependence on $n$ nearly matches the upper bound of [DOS17, NP17] when $\ell = O(1)$, and we reduce the dependence on $\ell$ from doubly to singly exponential. Therefore, we are able to learn large mixtures of strings: while Ban et al.'s algorithm can only learn a mixture of $O(\log n/\log \log n)$ strings with a subexponential number of samples, we are able to learn a mixture of $n^{o(1)}$ strings in $\exp\big(n^{1/3 + o(1)}\big)$ samples and time.

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。

摘要：

英文摘要：

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