Optimal rates for ranking a permuted isotonic matrix in polynomial time

TL;DR

This paper introduces a polynomial-time method for optimally recovering the permutation in a permuted isotonic matrix, improving ranking accuracy in models like crowd-labeling and tournament comparisons.

Contribution

It provides the first optimal polynomial-time algorithm for permutation recovery in permuted isotonic matrices, addressing an open problem and enhancing ranking methods in stochastic transitivity models.

Findings

Achieved optimal permutation recovery rates.

Improved ranking accuracy in stochastic transitivity models.

Developed new concentration inequalities for sparse matrices.

Abstract

We consider a ranking problem where we have noisy observations from a matrix with isotonic columns whose rows have been permuted by some permutation $\pi$ *. This encompasses many models, including crowd-labeling and ranking in tournaments by pair-wise comparisons. In this work, we provide an optimal and polynomial-time procedure for recovering $\pi$ * , settling an open problem in [7]. As a byproduct, our procedure is used to improve the state-of-the art for ranking problems in the stochastically transitive model (SST). Our approach is based on iterative pairwise comparisons by suitable data-driven weighted means of the columns. These weights are built using a combination of spectral methods with new dimension-reduction techniques. In order to deal with the important case of missing data, we establish a new concentration inequality for sparse and centered rectangular Wishart-type matrices.