Compressive sensing for Gauss-Gauss detection

The recently introduced theory of compressed sensing (CS) enables the reconstruction of sparse signals from a small set of linear measurements. If properly chosen, the number of measurements can be much smaller than the number of Nyquist rate samples. However, despite the intense focus on the reconstruction of signals, many signal processing problems do not require a full reconstruction of the signal and little attention has been paid to doing inference in the CS domain. In this paper we show the performance of CS for the problem of signal detection using Gauss-Gauss detection. We investigate how the J-divergence and Fisher Discriminant are affected when used in the CS domain. In particular, we demonstrate how to perform detection given the measurements without ever reconstructing the signals themselves and provide theoretical bounds on the performance. A numerical example is provided to demonstrate the effectiveness of CS under Gauss-Gauss detection.