In this paper, we study the minimax rates and provide an implementable convex algorithm for Poisson inverse problems under weak sparsity and physical constraints. In particular, we assume the model yi~ Poisson(TaiTf*) for 1 ≤ i ≤ n, where T ∈ ℝ+is the intensity, and we impose weak sparsity on f* ∈ ℝpby assuming f* lies in an ℓq-ball when rotated according to an orthonormal basis D ∈ ℝp×p. In addition, since we are modeling real-physical systems, we also impose positivity and flux-preserving constraints on the matrix A = [a1, a2, ... , an]Tand the function f*. We prove minimax lower bounds for this model, which scale as Rq(log p/T)1-(q/2)where it is noticeable that the rate depends on the intensity T and not the sample size n. We also show that an ℓ1-based regularized least-squares estimator achieves this minimax lower bound, provided a suitable restricted eigenvalue condition is satisfied. Finally, we prove that provided n ≥ K̃ log p where K̃ = Θ(Rq(log p/T)-(q/2)) represents an approximate sparsity level, and our restricted eigenvalue condition and physical constraints are satisfied for random bounded ensembles. We also provide numerical experiments that validate our mean-squared error bounds. Our results address a number of open issues from prior work on Poisson inverse problems that focuses on strictly sparse models and does not provide guarantees for convex implementable algorithms.