  In this paper, we investigate threshold effects associated with swapping of
signal and noise subspaces in estimating signal parameters from compressed
noisy data. The term threshold effect refers to a sharp departure of
mean-squared error from the Cramer-Rao bound when the signal-to-noise ratio
falls below a threshold SNR. In many cases, the threshold effect is caused by a
subspace swap event, when the measured data (or its sample covariance) is
better approximated by a subset of components of an orthogonal subspace than by
the components of a signal subspace. We derive analytical lower bounds on the
probability of a subspace swap in compressively measured noisy data. These
bounds guide our understanding of threshold effects and performance breakdown
for parameter estimation using compression. As a case study, we investigate
threshold effects in maximum likelihood (ML) estimation of directions of
arrival of two closely-spaced sources using co-prime subsampling. Our results
show the impact of compression on threshold SNR. A rule of thumb is that every
doubling of compression ratio brings a penalty in threshold SNR of 3 dB.
