  This paper derives fundamental limits on the performance of compressive
classification when the source is a mixture of Gaussians. It provides an
asymptotic analysis of a Bhattacharya based upper bound on the
misclassification probability for the optimal Maximum-A-Posteriori (MAP)
classifier that depends on quantities that are dual to the concepts of
diversity-order and coding gain in multi-antenna communications. The
diversity-order of the measurement system determines the rate at which the
probability of misclassification decays with signal-to-noise ratio (SNR) in the
low-noise regime. The counterpart of coding gain is the measurement gain which
determines the power offset of the probability of misclassification in the
low-noise regime. These two quantities make it possible to quantify differences
in misclassification probability between random measurement and
(diversity-order) optimized measurement. Results are presented for two-class
classification problems first with zero-mean Gaussians then with nonzero-mean
Gaussians, and finally for multiple-class Gaussian classification problems. The
behavior of misclassification probability is revealed to be intimately related
to certain fundamental geometric quantities determined by the measurement
system, the source and their interplay. Numerical results, representative of
compressive classification of a mixture of Gaussians, demonstrate alignment of
the actual misclassification probability with the Bhattacharya based upper
bound. The connection between the misclassification performance and the
alignment between source and measurement geometry may be used to guide the
design of dictionaries for compressive classification.
