General regularization schemes for signal detection in inverse problems

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 a test which adapts (up to a log log factor) to the unknown smoothness in the alternative. Moreover, the authors discuss how the so-called direct and indirect tests are related in terms of interpolation properties.