Random distortion testing with linear measurements

This paper addresses the problem of testing whether, after linear transformations and possible dimensionality reductions, a random matrix of interest Θ deviates significantly from some matrix model θ0, when Θ is observed in additive independent Gaussian noise with known covariance matrix. In contrast to standard likelihood theory, the probability distribution of Θ is assumed to be unknown. This problem generalizes the Random Distortion Testing (RDT) problem addressed in a former paper. Although the notions of size and power can be extended so as to deal with this generalized problem, no Uniformly Most Powerful (UMP) test exists for it. We can however exhibit a relevant subclass of tests and prove the existence of an optimal test within this class, that is, a test with specified size and maximal constant conditional power. As a consequence, this test is also UMP among invariant tests. The method fits within a wide range of signal processing scenarios. It is here specifically applied to sequential detection, subspace detection and random distortion testing after space-time compressive sensing. It is also used to extend the GLRT optimality properties for testing a waveform amplitude in noise.