  We study the rate of decay of the probability of error for distinguishing
between a sparse signal with noise, modeled as a sparse mixture, from pure
noise. This problem has many applications in signal processing, evolutionary
biology, bioinformatics, astrophysics and feature selection for machine
learning. We let the mixture probability tend to zero as the number of
observations tends to infinity and derive oracle rates at which the error
probability can be driven to zero for a general class of signal and noise
distributions via the likelihood ratio test. In contrast to the problem of
detection of non-sparse signals, we see the log-probability of error decays
sublinearly rather than linearly and is characterized through the
$\chi^2$-divergence rather than the Kullback-Leibler divergence for "weak"
signals and can be independent of divergence for "strong" signals. Our
contribution is the first characterization of the rate of decay of the error
probability for this problem for both the false alarm and miss probabilities.
