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,我们有一个具有未知排列$\pi$*的双变量等保n x d矩阵的部分观测值。针对恢复排列$\pi$*和估计未知矩阵这两个问题,我们引入了一种多项式时间的程序,该程序在所有可能的n、d值和所有可能的采样努力下实现了这两个问题的最小风险。其中,我们确立了在某些情况下,恢复未知排列$\pi$*比估计矩阵要简单得多。