  Crowdsourcing is a strategy to categorize data through the contribution of
many individuals. A wide range of theoretical and algorithmic contributions are
based on the model of Dawid and Skene [1]. Recently it was shown in [2,3] that,
in certain regimes, belief propagation is asymptotically optimal for data
generated from the Dawid-Skene model. This paper is motivated by this recent
progress. We analyze the dense limit of the Dawid-Skene model. It is shown that
it belongs to a larger class of low-rank matrix estimation problems for which
it is possible to express the asymptotic, Bayes-optimal, performance in a
simple closed form. In the dense limit the mapping to a low-rank matrix
estimation problem provides an approximate message passing algorithm that
solves the problem algorithmically. We identify the regions where the algorithm
efficiently computes the Bayes-optimal estimates. Our analysis refines the
results of [2,3] about optimality of message passing algorithms by
characterizing regions of parameters where these algorithms do not match the
Bayes-optimal performance. We further study numerically the performance of
approximate message passing, derived in the dense limit, on sparse instances
and carry out experiments on a real world dataset.
