Detection with phaseless measurements

We consider the problem of hypothesis testing for detection of a signal in Gaussian noise. We assume that the vector of measurements is unobserved, and that our observations consist of phaseless inner products with a set of known measurement vectors. This is typical of the phase retrieval problem, where the goal is to recover the vector of measurements. We provide a simple estimator for the test statistic that does not necessitate a phaseless recovery method to reconstruct the measurements. Our analysis shows that for random measurement vectors, we can reconstruct the test statistic for any signal from a sufficient number of observations, quadratic in the signal length, using a simple least-squares approach. The primary advantage of this method its simplicity and computational efficiency, which comes at the expense of requiring many more measurements. We show that for Fourier measurements vectors, our approach works only when the signal is also a Fourier vector.