Optimal Permutation Estimation in CrowdSourcing problems

Abstract

Motivated by crowdsourcing applications, we consider a model where we have partial observations from a bivariate isotonic $\mathit{n}\times \mathit{d}$ matrix with an unknown permutation ${\mathit{\pi }}^{\ast }$ acting on its rows. Focusing on the twin problems of recovering the permutation ${\mathit{\pi }}^{\ast }$ 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 ${\mathit{\pi }}^{\ast }$ is considerably simpler than estimating the matrix.