@article{CarpentierNickl2015,
  author    = {Carpentier, Alexandra and Nickl, Richard},
  title     = {On signal detection and confidence sets for low rank inference problems},
  series = {Electronic journal of statistics},
  volume    = {9},
  journal   = {Electronic journal of statistics},
  number    = {2},
  publisher = {Institute of Mathematical Statistics},
  address   = {Cleveland},
  issn      = {1935-7524},
  doi       = {10.1214/15-EJS1087},
  pages     = {2675 -- 2688},
  year      = {2015},
  abstract  = {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 [5] for confidence sets in low rank recovery problems are essentially optimal.},
  language  = {en}
}
@article{CarpentierKloppLoeffleretal.2018,
  author    = {Carpentier, Alexandra and Klopp, Olga and L{\"o}ffler, Matthias and Nickl, Richard},
  title     = {Adaptive confidence sets for matrix completion},
  series = {Bernoulli : official journal of the Bernoulli Society for Mathematical Statistics and Probability},
  volume    = {24},
  journal   = {Bernoulli : official journal of the Bernoulli Society for Mathematical Statistics and Probability},
  number    = {4A},
  publisher = {International Statistical Institute},
  address   = {Voorburg},
  issn      = {1350-7265},
  doi       = {10.3150/17-BEJ933},
  pages     = {2429 -- 2460},
  year      = {2018},
  abstract  = {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.},
  language  = {en}
}