打破$1/\sqrt{n}$障碍：多项式时间中基于置换的模型的更快速率
Breaking the $1/\sqrt{n}$ Barrier: Faster Rates for Permutation-based Models in Polynomial Time
摘要：
许多应用，包括秩聚合和群体标记，可以建模的二元保序矩阵与未知的排列作用在其行和列。我们考虑的问题，估计这样的矩阵的基础上的一个子集的条目的噪声观测，并设计和分析一个多项式时间的算法，提高了最先进的状态。特别是，我们的结果意味着，任何这样的$n \times n$矩阵可以有效地估计在归一化Frobenius范数的速率$\widetilde{\mathcal O}(n^{-3/4})$，从而缩小$\widetilde{\mathcal O}(n^{-1})$和$\widetilde{\mathcal O}(n^{-1/2})$之间的差距，这分别是迄今为止统计和计算效率最高的方法的比率。
英文摘要：
Many applications, including rank aggregation and crowd-labeling, can be modeled in terms of a bivariate isotonic matrix with unknown permutations acting on its rows and columns. We consider the problem of estimating such a matrix based on noisy observations of a subset of its entries, and design and analyze a polynomial-time algorithm that improves upon the state of the art. In particular, our results imply that any such $n \times n$ matrix can be estimated efficiently in the normalized Frobenius norm at rate $\widetilde{\mathcal O}(n^{-3/4})$, thus narrowing the gap between $\widetilde{\mathcal O}(n^{-1})$ and $\widetilde{\mathcal O}(n^{-1/2})$, which were hitherto the rates of the most statistically and computationally efficient methods, respectively.
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。
