# Breaking the $1/\sqrt{n}$ Barrier: Faster Rates for Permutation-based Models in Polynomial Time

## TL;DR

This paper introduces a polynomial-time algorithm for estimating permutation-based isotonic matrices that achieves faster convergence rates, narrowing the gap between statistical and computational efficiency in matrix estimation.

## Contribution

The authors develop a new algorithm that improves the estimation rate for permutation-based models, achieving a rate of O(n^{-3/4}) in polynomial time, surpassing previous methods.

## Key findings

- Achieves O(n^{-3/4}) rate in Frobenius norm
- Efficient polynomial-time algorithm for permutation-based matrix estimation
- Narrowing the gap between statistical and computational rates

## Abstract

Many applications, including rank aggregation and crowd-labeling, can be modeled in terms of a bivariate isotonic matrix with unknown permutations acting on its rows and columns. We consider the problem of estimating such a matrix based on noisy observations of a subset of its entries, and design and analyze a polynomial-time algorithm that improves upon the state of the art. In particular, our results imply that any such $n \times n$ matrix can be estimated efficiently in the normalized Frobenius norm at rate $\widetilde{\mathcal O}(n^{-3/4})$, thus narrowing the gap between $\widetilde{\mathcal O}(n^{-1})$ and $\widetilde{\mathcal O}(n^{-1/2})$, which were hitherto the rates of the most statistically and computationally efficient methods, respectively.