As motivation, we consider a bunch of experts responding to a set of questions, where we can observe whether an expert answers a question correctly or not. Assume that for every pair of experts, one of the experts has for every question at least the same probability to answer correctly as the other expert. This means that they can be ordered by quality, and moreover we assume that the questions can be ordered by difficulty in the same sense. Storing the probabilities of a correct answer yields a matrix, where each row corresponds to an expert and each column corresponds to a question. By assumption, this matrix can be ordered such that each row and column is non-decreasing. We call such a matrix bi-isotonic.
More general, we consider the observation model of an underlying permuted bi-isotonic matrix with sub-Gaussian noise. The aim of the talk is to give an overview over existing results and bounds for estimating the underlying matrix in different settings. We will see that estimating the respective permutations is driving the error in most regimes. At the end, we will focus on the special case, where the underlying matrix only takes two values.
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