Nonconvex relaxation for Poisson intensity reconstruction

Critical to accurate reconstruction of sparse signals from low-dimensional Poisson observations is the solution of nonlinear optimization problems that promote sparse solutions. Theoretically, non-convex ℓ_p-norm minimization (0 ≤ p <; 1) would lead to more accurate reconstruction than the convex ℓ₁-norm relaxation commonly used in sparse signal recovery. In this paper, we propose an extension to the existing SPIRAL-ℓ₁ algorithm based on the Generalized Soft-Thersholding (GST) function to better recover signals with mostly nonzero entries from Poisson observations. This approach is based on iteratively minimizing a sequence of separable subproblems of the nonnegatively constrained, ℓ_p-penalized negative Poisson log-likelihood objective function using the GST function. We demonstrate the effectiveness of the proposed method, called SPIRAL-ℓ_p, through numerical experiments.