3. Sparse Signal Recovery with Exponential-Family Noise

The problem of sparse signal recovery from a relatively small number of noisy measurements has been studied extensively in the recent compressed sensing literature. Typically, the signal reconstruction problem is formulated as \(l_1\)-regularized linear regression. From a statistical point of view, this problem is equivalent to maximum a posteriori probability (MAP) parameter estimation with Laplace prior on the vector of parameters (i.e., signal) and linear measurements disturbed by Gaussian noise. Classical results in compressed sensing (e.g., [7]) state sufficient conditions for accurate recovery of noisy signals in such linear-regression setting. A natural question to ask is whether one can accurately recover sparse signals under different noise assumptions. Herein, we extend the results of [7] to the general case of exponential-family noise that includes Gaussian noise as a particular case; the recovery problem is then formulated as \(l_1\)-regularized Generalized Linear Model (GLM) regression. We show that, under standard restricted isometry property (RIP) assumptions on the design matrix, \(l_1\)-minimization can provide stable recovery of a sparse signal in presence of exponential-family noise, and state some sufficient conditions on the noise distribution that guarantee such recovery.