On signal detection and confidence sets for low rank inference problems

TL;DR

This paper establishes the exact detection boundaries for low rank matrices in Gaussian trace regression and demonstrates the fundamental limitations in constructing adaptive confidence sets for such matrices.

Contribution

It derives the precise detection boundaries for low rank signals and proves the non-existence of adaptive confidence sets in nuclear norm, confirming the optimality of prior positive results.

Findings

Exact detection boundaries for Frobenius and nuclear norms.

Non-existence of adaptive confidence sets for low rank matrices.

Validation of previous positive results as essentially optimal.

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 (Carpentier, Eisert, Gross and Nickl, 2015) for confidence sets in low rank recovery problems are essentially optimal.