Threshold effects in parameter estimation from compressed data

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 catastrophic increase in mean-squared error 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 the 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 MUSIC estimation of direction of arrivals of two closely-spaced sources using Gaussian random compression and co-prime subsampling. Our results show the impact of compression on threshold SNR.