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Many applications, including rank aggregation and crowd-labeling, can bemodeled in terms of a bivariate isotonic matrix with unknown permutationsacting on its rows and columns. We consider the problem of estimating such amatrix based on noisy observations of a subset of its entries, and design andanalyze a polynomial-time algorithm that improves upon the state of the art. Inparticular, our results imply that any such $n \times n$ matrix can beestimated 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 $\widetilde{\mathcal O}(n^{-1/2})$, whichwere hitherto the rates of the most statistically and computationally efficientmethods, respectively.

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