We consider the problem of sparsity pattern detection for unknown
$k$-sparse $n$-dimensional signals observed through $m$ noisy, random
linear measurements.  Our analyses are for asymptotically-reliable
sparsity pattern recovery in terms of the dimensions $m$, $n$ and $k$
as well as the signal-to-noise ratio (SNR) and the minimum-to-average
ratio (MAR) of the nonzero entries of the signal.  With a new
necessary condition for the success of any recovery algorithm and
a new sufficient condition for the success of a simple thresholding
algorithm, we are able to explain the relative performance of
thresholding, lasso, and maximum likelihood (ML) estimation.
Thresholding requires 4(1+SNR) times more measurements than ML and
4/MAR times more measurements than lasso.  This provides insight on
the precise value and limitations of sparsity pattern detection
algorithms.