A consistent and numerically efficient variable selection method for sparse Poisson regression with applications to learning and signal recovery

We propose an adaptive 1-penalized estimator in the framework of Generalized Linear Models with identity-link and Poisson data, by taking advantage of a globally quadratic approximation of the Kullback-Leibler divergence. We prove that this approximation is asymptotically unbiased and that the proposed estimator has the variable selection consistency property in a deterministic matrix design framework. Moreover, we present a numerically efficient strategy for the computation of the proposed estimator, making it suitable for the analysis of massive counts datasets. We show with two numerical experiments that the method can be applied both to statistical learning and signal recovery problems.