  This paper considers fundamental limits for solving sparse inverse problems
in the presence of Poisson noise with physical constraints. Such problems arise
in a variety of applications, including photon-limited imaging systems based on
compressed sensing. Most prior theoretical results in compressed sensing and
related 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 and
physical constraints provided upper bounds on mean squared error performance
for a specific class of estimators. However, it was unknown whether those
bounds were tight or if other estimators could achieve significantly better
performance. This work provides minimax lower bounds on mean-squared error for
sparse Poisson inverse problems under physical constraints. Our lower bounds
are complemented by minimax upper bounds. Our upper and lower bounds reveal
that due to the interplay between the Poisson noise model, the sparsity
constraint and the physical constraints: (i) the mean-squared error does not
depend on the sample size $n$ other than to ensure the sensing matrix satisfies
RIP-like conditions and the intensity $T$ of the input signal plays a critical
role; and (ii) the mean-squared error has two distinct regimes, a low-intensity
and a high-intensity regime and the transition point from the low-intensity to
high-intensity regime depends on the input signal $f^*$. In the low-intensity
regime the mean-squared error is independent of $T$ while in the high-intensity
regime, the mean-squared error scales as $\frac{s \log p}{T}$, where $s$ is the
sparsity level, $p$ is the number of pixels or parameters and $T$ is the signal
intensity.
