Motivated by crowd-sourcing applications, we consider a model where we have partial observations from a bivariate isotonic n x d matrix with an unknown permutation $\pi$ * acting on its rows. Focusing on the twin problems of recovering the permutation $\pi$ * and estimating the unknown matrix, we introduce a polynomial-time procedure achieving the minimax risk for these two problems, this for all possible values of n, d, and all possible sampling efforts. Along the way, we establish that, in some regimes, recovering the unknown permutation $\pi$ * is considerably simpler than estimating the matrix.

翻译：受众包应用启发，我们考虑一个模型，其中我们从具有未知排列 $\pi$ *（作用于行）的双变量单调 n x d 矩阵中获取部分观测值。针对恢复排列 $\pi$ * 和估计未知矩阵这两个相伴问题，我们引入了一种多项式时间程序，该程序在所有可能的 n、d 值及所有可能的采样力度下，均能达到这两个问题的极小化极大风险。在此过程中，我们发现在某些情形下，恢复未知排列 $\pi$ * 比估计矩阵要简单得多。