Breaking the 1/√(n) Barrier: Faster Rates for Permutation-based Models in Polynomial Time
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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 polynomial-time algorithms that improve upon the state of the art. In particular, our results imply that any such n × n matrix can be estimated efficiently in the normalized Frobenius norm at rate O(n^-3/4), thus narrowing the gap between O(n^-1) and O(n^-1/2), which were hitherto the rates of the most statistically and computationally efficient methods, respectively.
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