  This paper studies the classification of high-dimensional Gaussian signals
from low-dimensional noisy, linear measurements. In particular, it provides
upper bounds (sufficient conditions) on the number of measurements required to
drive the probability of misclassification to zero in the low-noise regime,
both for random measurements and designed ones. Such bounds reveal two
important operational regimes that are a function of the characteristics of the
source: i) when the number of classes is less than or equal to the dimension of
the space spanned by signals in each class, reliable classification is possible
in the low-noise regime by using a one-vs-all measurement design; ii) when the
dimension of the spaces spanned by signals in each class is lower than the
number of classes, reliable classification is guaranteed in the low-noise
regime by using a simple random measurement design. Simulation results both
with synthetic and real data show that our analysis is sharp, in the sense that
it is able to gauge the number of measurements required to drive the
misclassification probability to zero in the low-noise regime.
