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  - 
TY  - JOUR
A1  - Carpentier, Alexandra
A1  - Kim, Arlene K. H.
T1  - An iterative hard thresholding estimator for low rank matrix recovery with explicit limiting distribution
JF  - Statistica Sinica
N2  - We consider the problem of low rank matrix recovery in a stochastically noisy high-dimensional setting. We propose a new estimator for the low rank matrix, based on the iterative hard thresholding method, that is computationally efficient and simple. We prove that our estimator is optimal in terms of the Frobenius risk and in terms of the entry-wise risk uniformly over any change of orthonormal basis, allowing us to provide the limiting distribution of the estimator. When the design is Gaussian, we prove that the entry-wise bias of the limiting distribution of the estimator is small, which is of interest for constructing tests and confidence sets for low-dimensional subsets of entries of the low rank matrix.
KW  - High dimensional statistical inference
KW  - inverse problem
KW  - limiting distribution
KW  - low rank matrix recovery
KW  - numerical methods
KW  - uncertainty quantification
Y1  - 2018
U6  - https://doi.org/10.5705/ss.202016.0103
SN  - 1017-0405
SN  - 1996-8507
VL  - 28
IS  - 3
SP  - 1371
EP  - 1393
PB  - Statistica Sinica, Institute of Statistical Science, Academia Sinica
CY  - Taipei
ER  - 
TY  - JOUR
A1  - Blanchard, Gilles
A1  - Carpentier, Alexandra
A1  - Gutzeit, Maurilio
T1  - Minimax Euclidean separation rates for testing convex hypotheses in R-d
JF  - Electronic journal of statistics
N2  - We consider composite-composite testing problems for the expectation in the Gaussian sequence model where the null hypothesis corresponds to a closed convex subset C of R-d. We adopt a minimax point of view and our primary objective is to describe the smallest Euclidean distance between the null and alternative hypotheses such that there is a test with small total error probability. In particular, we focus on the dependence of this distance on the dimension d and variance 1/n giving rise to the minimax separation rate. In this paper we discuss lower and upper bounds on this rate for different smooth and non-smooth choices for C.
KW  - Minimax hypothesis testing
KW  - Gaussian sequence model
KW  - nonasymptotic minimax separation rate
Y1  - 2018
U6  - https://doi.org/10.1214/18-EJS1472
SN  - 1935-7524
VL  - 12
IS  - 2
SP  - 3713
EP  - 3735
PB  - Institute of Mathematical Statistics
CY  - Cleveland
ER  -