Optimal Permutation Estimation in Crowd-Sourcing problems

TL;DR

This paper introduces a polynomial-time method for accurately recovering an unknown permutation and estimating a bivariate isotonic matrix in crowd-sourcing, demonstrating that permutation recovery can be easier than matrix estimation in certain regimes.

Contribution

It presents a novel polynomial-time procedure that achieves minimax risk for permutation recovery and matrix estimation in crowd-sourcing models, applicable across all data sizes and sampling efforts.

Findings

The method achieves minimax optimality in permutation and matrix estimation.

Permutation recovery can be easier than matrix estimation in some regimes.

The approach is effective for all sample sizes and sampling efforts.

Abstract

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.