Minimax optimal convex methods for Poisson inverse problems under $\ell_q$-ball sparsity

In this paper, we study the minimax rates and provide a convex implementable 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 convex $\ell_1$-based regularized least-squares estimator achieves this minimax lower bound up to a $\log n$ factor, provided a suitable restricted eigenvalue condition is satisfied. Finally we prove that provided $n$ is sufficiently large, our restricted eigenvalue condition and physical constraints are satisfied for random bounded ensembles. Our results address a number of open issues from prior work on Poisson inverse problems that focusses on strictly sparse models and does not provide guarantees for convex implementable algorithms.