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 literature on compressed sensing. However, the focus of those studies appears to be limited to the case of linear projections disturbed by Gaussian noise, and the sparse signal reconstruction problem is treated as linear regression with l₁-norm regularization constraint. A natural question to ask is whether one can accurately recover sparse signals under different noise assumptions. Herein, we extend the results of to the more general case of exponential-family noise that includes Gaussian noise as a particular case, and yields l₁-regularized generalized linear model (GLM) regression problem. We show that, under standard restricted isometry property (RIP) assumptions on the design matrix, l₁-minimization can provide stable recovery of a sparse signal in presence of the exponential-family noise, provided that certain sufficient conditions on the noise distribution are satisfied.