  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 $y_i \sim
\mbox{Poisson}(Ta_i^{\top}f^*)$ for $1 \leq i \leq n$ where $T \in
\mathbb{R}_+$ is the intensity, and we impose weak sparsity on $f^* \in
\mathbb{R}^p$ by assuming $f^*$ lies in an $\ell_q$-ball when rotated according
to an orthonormal basis $D \in \mathbb{R}^{p \times p}$. In addition, since we
are modeling real physical systems we also impose positivity and
flux-preserving constraints on the matrix $A = [a_1, a_2,...,a_n]^{\top}$ and
the function $f^*$. We prove minimax lower bounds for this model which scale as
$R_q (\frac{\log p}{T})^{1 - q/2}$ where it is noticeable that the rate depends
on the intensity $T$ and not the sample size $n$. We also show that a
$\ell_1$-based regularized least-squares estimator achieves this minimax lower
bound, provided a suitable restricted eigenvalue condition is satisfied.
Finally we prove that provided $n \geq \tilde{K} \log p$ where $\tilde{K} =
O(R_q (\frac{\log p}{T})^{- q/2})$ represents an approximate sparsity level,
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.
