Minimax Optimal Rates for Poisson Inverse Problems With Physical Constraints

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 (CS). Most prior theoretical results in CS and related inverse problems apply to idealized settings where the noise is independent identically distributed and do not account for signal-dependent noise and physical sensing constraints. Prior results on Poisson CS with signal-dependent noise and physical constraints provided upper bounds on mean-squared error (MSE) 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 paper provides minimax lower bounds on MSE for sparse Poisson inverse problems under physical constraints. The lower bounds are complemented by minimax upper bounds which match the lower bounds for certain problem sizes and noise levels. The source of the mismatch between upper and lower bounds for other problem sizes and noise levels is discussed. The upper and lower bounds reveal that due to the interplay between the Poisson noise model, the sparsity constraint and the physical constraints: 1) the MSE upper bound does not depend on the sample size n other than to ensure the sensing matrix satisfies Restricted Isometry Property-like conditions and the intensity T of the input signal plays a critical role and 2) the MSE upper bound has two distinct regimes, corresponding to low and high intensities, and the transition point from the low-intensity to high-intensity regime depends on the sparsifying basis D. In the low-intensity regime, the MSE upper bound is independent of T while in the high-intensity regime, the MSE upper bound scales as (slog p/T), where s is the sparsity level, p is the number of pixels or parameters, and T is the signal intensity.