General regularization schemes for signal detection in inverse problems

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.