  In the present paper we study the problem of existence of honest and adaptive
confidence sets for matrix completion. We consider two statistical models: the
trace regression model and the Bernoulli model. In the trace regression model,
we show that honest confidence sets that adapt to the unknown rank of the
matrix exist even when the error variance is unknown. Contrary to this, we
prove that in the Bernoulli model, honest and adaptive confidence sets exist
only when the error variance is known a priori. In the course of our proofs we
obtain bounds for the minimax rates of certain composite hypothesis testing
problems arising in low rank inference.
