Minimax Optimal Convex Methods for Poisson Inverse Problems Under <inline-formula> <tex-math notation="LaTeX">$\ell_{q}$ </tex-math> </inline-formula>-Ball Sparsity

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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 $y_{i} \sim \mathrm {Poisson}(Ta_{i}^{\top }f^{*})$ 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 $A = [a_{1}, a_{2},\ldots, a_{n}]^{\top }$ and the function $f^{*}$ . We prove minimax lower bounds for this model, which scale as $R_{q} ({\log p}/{T})^{1 - ({q}/{2})}$ where it is noticeable that the rate depends on the intensity $T$ and not the sample size $n$ . We also show that an $\ell _{1}$ -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 $\tilde {K} = \Theta (R_{q} ({\log p}/{T})^{- ({q}/{2})})$ 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.