Towards Optimal Estimation of Bivariate Isotonic Matrices with Unknown Permutations
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Many applications, including rank aggregation, crowd-labeling, and graphon estimation, 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, squared Frobenius norm at rate O(n^-3/4), thus narrowing the gap between O(n^-1) and O(n^-1/2), hitherto the rates of the most statistically and computationally efficient methods, respectively. Additionally, our algorithms are minimax optimal in another natural metric that measures how well an estimate captures each row of the matrix. Along the way, we prove matching upper and lower bounds on the minimax radii of certain cone testing problems, which may be of independent interest.
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