Threshold Effects in Parameter Estimation From Compressed Data

In this paper, we investigate threshold effects associated with the 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 Cramér-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 in two canonical models: a first-order model and a second-order model. In the first-order model, the parameters to be estimated modulate the mean of a complex multivariate normal set of measurements. In the second-order model, the parameters modulate the covariance of complex multivariate measurements. In both cases, the probability bounds are tail probabilities of F-distributions, and they apply to any linear compression scheme. These lower bounds guide our understanding of threshold effects and performance breakdowns for parameter estimation using compression. In particular, they can be used to quantify the increase in threshold SNR as a function of a compression ratio C. We demonstrate numerically that this increase in threshold SNR is roughly 10log10 C dB, which is consistent with the performance loss that one would expect when measurements in Gaussian noise are compressed by a factor C.