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：受众包应用的启发，我们考虑一个模型：在该模型中，我们从一个具有未知行置换π*的双变量等序n×d矩阵中获取部分观测值。聚焦于恢复置换π*和估计未知矩阵这两个相关的问题，我们引入了一种多项式时间算法，该算法在n、d的所有可能取值以及所有采样努力程度下，均能实现这两个问题的最小化最大风险。在此过程中，我们证明了在某些情况下，恢复未知置换π*比估计矩阵要简单得多。