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：摘要：受众包应用启发，我们考虑了一个模型，其中我们有一个含有未知排列$\pi$* 的双变量等保 n x d 矩阵的部分观测。针对恢复排列$\pi$* 和估计未知矩阵的问题，我们提出了一个多项式时间的程序，使得在所有可能的 n、d 值和所有可能的采样努力下，这两个问题的最小化风险达到极值。在此过程中，我们确定了一些区域，在这些区域内，恢复未知排列$\pi$* 要比估计矩阵简单得多。