Towards Optimal Estimation of Bivariate Isotonic Matrices with Unknown Permutations

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/or columns. We consider the problem of estimating an unknown matrix in this class, based on noisy observations of (possibly, a subset of) its entries. We design and analyze polynomial-time algorithms that improve upon the state of the art in two distinct metrics, showing, in particular, that minimax optimal, computationally efficient estimation is achievable in certain settings. 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.