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This paper considers fundamental limits for solving sparse inverse problemsin the presence of Poisson noise with physical constraints. Such problems arisein a variety of applications, including photon-limited imaging systems based oncompressed sensing. Most prior theoretical results in compressed sensing andrelated inverse problems apply to idealized settings where the noise is i.i.d.,and do not account for signal-dependent noise and physical sensing constraints.Prior results on Poisson compressed sensing with signal-dependent noise andphysical constraints provided upper bounds on mean squared error performancefor a specific class of estimators. However, it was unknown whether thosebounds were tight or if other estimators could achieve significantly betterperformance. This work provides minimax lower bounds on mean-squared error forsparse Poisson inverse problems under physical constraints. Our lower boundsare complemented by minimax upper bounds. Our upper and lower bounds revealthat due to the interplay between the Poisson noise model, the sparsityconstraint and the physical constraints: (i) the mean-squared error does notdepend on the sample size $n$ other than to ensure the sensing matrix satisfiesRIP-like conditions and the intensity $T$ of the input signal plays a criticalrole; and (ii) the mean-squared error has two distinct regimes, a low-intensityand a high-intensity regime and the transition point from the low-intensity tohigh-intensity regime depends on the input signal $f^*$. In the low-intensityregime the mean-squared error is independent of $T$ while in the high-intensityregime, the mean-squared error scales as $\frac{s \log p}{T}$, where $s$ is thesparsity level, $p$ is the number of pixels or parameters and $T$ is the signalintensity.

Minimax Optimal Rates for Poisson Inverse Problems with Physical Constraints