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  for confidence sets in low rank recovery problems are essentially optimal.
"On signal detection and confidence sets for low rank inference problems." Electron. J. Statist. 9 (2) 2675 - 2688, 2015. https://doi.org/10.1214/15-EJS1087