Adaptive confidence sets for matrix completion

TL;DR

This paper investigates the existence of honest and adaptive confidence sets for matrix completion under two models, revealing conditions under which such sets can be constructed even with unknown error variance.

Contribution

It establishes the existence of adaptive confidence sets in the trace regression model without known error variance, and highlights limitations in the Bernoulli model.

Findings

Adaptive confidence sets exist in the trace regression model with unknown variance.

Such confidence sets do not exist in the Bernoulli model unless the variance is known.

Provides bounds for minimax rates in low rank hypothesis testing.

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.