Adaptive confidence sets for matrix completion

Alexandra Carpentier (1) , Olga Klopp (2, 3) , Matthias Löffler (4) , Richard Nickl (4)

1 University of Potsdam = Universität Potsdam
2 CREST - Centre de Recherche en Économie et Statistique
3 MODAL'X - Modélisation aléatoire de Paris X
4 CAM - University of Cambridge [UK]

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.

Mots clés: Low rank recovery; confidence sets; adaptivity; matrix completion; unknown variance; minimax hypothesis testing

Domaines: Statistiques [math.ST]

Alexandra Carpentier, Olga Klopp, Matthias Löffler, Richard Nickl. Adaptive confidence sets for matrix completion. Bernoulli, 2018, 24 (4A), pp.2429 - 2460. ⟨hal-01354030v2⟩