Canonical Coordinates are the Right Coordinates for Low-Rank Gauss–Gauss Detection and Estimation

In this correspondence, our aim is to establish a connection between low-rank detection, low-rank estimation, and canonical coordinates. The key to this connection is the observation that Gauss-Gauss detectors and estimators share canonical coordinates in the case where the underlying model is a signal-plus-noise model. We show that in Gauss-Gauss detection J-divergence is a function of squared canonical correlations, and hence is invariant to nonsingular transformations of the data channels. Further, we show that J-divergence has a special decomposition in canonical coordinates, impelling their use for rank-reduction. Canonical coordinates have been found earlier to be fundamental for low-rank estimation, as they decompose three important performance measures, namely the relative volume of error concentration ellipse, processing gain, and information rate. This correspondence shows that canonical coordinates are also fundamental for low-rank detection, making them more useful in signal processing and communication problems when low-rank modeling is required to achieve computational efficiency or robustness against noise and model uncertainties