  We consider the signal detection problem in the Gaussian design trace
regression model with low rank alternative hypotheses. We derive the precise
(Ingster-type) detection boundary for the Frobenius and the nuclear norm. We
then apply these results to show that honest confidence sets for the unknown
matrix parameter that adapt to all low rank sub-models in nuclear norm do not
exist. This shows that recently obtained positive results in (Carpentier,
Eisert, Gross and Nickl, 2015) for confidence sets in low rank recovery
problems are essentially optimal.
