TY  - JOUR
A1  - Carpentier, Alexandra
A1  - Nickl, Richard
T1  - On signal detection and confidence sets for low rank inference problems
JF  - Electronic journal of statistics
N2  - 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.
KW  - Low rank matrices
KW  - confidence sets
KW  - signal detection
KW  - nuclear norm
Y1  - 2015
U6  - https://doi.org/10.1214/15-EJS1087
SN  - 1935-7524
VL  - 9
IS  - 2
SP  - 2675
EP  - 2688
PB  - Institute of Mathematical Statistics
CY  - Cleveland
ER  - 
TY  - JOUR
A1  - Carpentier, Alexandra
A1  - Klopp, Olga
A1  - Löffler, Matthias
A1  - Nickl, Richard
T1  - Adaptive confidence sets for matrix completion
JF  - Bernoulli : official journal of the Bernoulli Society for Mathematical Statistics and Probability
N2  - 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.
KW  - adaptivity
KW  - confidence sets
KW  - low rank recovery
KW  - matrix completion
KW  - minimax hypothesis testing
KW  - unknown variance
Y1  - 2018
U6  - https://doi.org/10.3150/17-BEJ933
SN  - 1350-7265
SN  - 1573-9759
VL  - 24
IS  - 4A
SP  - 2429
EP  - 2460
PB  - International Statistical Institute
CY  - Voorburg
ER  -