Low rank detectors for Gaussian random vectors

A constructive procedure is presented for designing low rank detectors which maximize the divergence between hypotheses about the covariance structure of Gaussian signals. The detectors are constructed from the eigenstructure of a "signal-to-noise ratio" matrix. We show that the unity eigenvalues of this matrix provide no information about the covariance structure and the rank of the detector may be reduced with no performance penalty. More generally, we show that in certain situations, small eigenvalues may provide more information for discrimination than large eigenvalues. This leads to a systematic procedure for reducing the rank of a quadratic detector. Several examples are discussed.