General regularization schemes for signal detection in inverse problems

TL;DR

This paper explores general regularization methods for designing optimal signal detection tests in inverse problems, achieving minimax rates and adaptive procedures for unknown smoothness.

Contribution

It introduces a unified framework for linear and projection regularization schemes to develop minimax optimal and adaptive signal detection tests in inverse problems.

Findings

Regularization-based tests attain minimax separation rates.

Adaptive tests are developed that handle unknown smoothness.

Connections between direct and indirect tests via interpolation are established.

Abstract

The authors discuss how general regularization schemes, in particular linear regularization schemes and projection schemes, can be used to design tests for signal detection in statistical inverse problems. It is shown that such tests can attain the minimax separation rates when the regularization parameter is chosen appropriately. It is also shown how to modify these tests in order to obtain (up to a $\log\log$ factor) a test which adapts to the unknown smoothness in the alternative. Moreover, the authors discuss how the so-called \emph{direct} and \emph{indirect} tests are related via interpolation properties.