Breaking the $1/\sqrt{n}$ Barrier: Faster Rates for Permutation-based Models in Polynomial Time
This work addresses a computational bottleneck in applications like rank aggregation and crowd-labeling by providing a faster polynomial-time algorithm for matrix estimation, though it is incremental in narrowing the gap between existing rates.
The paper tackles the problem of estimating a bivariate isotonic matrix with unknown permutations from noisy observations, designing a polynomial-time algorithm that achieves a rate of $\widetilde{\mathcal O}(n^{-3/4})$ in normalized Frobenius norm, improving upon previous rates of $\widetilde{\mathcal O}(n^{-1})$ for statistical efficiency and $\widetilde{\mathcal O}(n^{-1/2})$ for computational efficiency.
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.