Reconstruction Error Bounds for Compressed Sensing under Poisson Noise using the Square Root of the Jensen-Shannon Divergence

Reconstruction error bounds in compressed sensing under Gaussian or uniform bounded noise do not translate easily to the case of Poisson noise. Reasons for this include the signal dependent nature of Poisson noise, and also the fact that the negative log likelihood in case of a Poisson distribution (which is directly related to the generalized Kullback-Leibler divergence) is not a metric and does not obey the triangle inequality. In this paper, we develop new performance error bounds for a computationally tractable estimator, for reconstruction from compressed measurements affected by Poisson noise, by replacing the generalized Kullback-Leibler divergence with an information theoretic metric namely the square root of the Jensen-Shannon divergence, which is related to a symmetrized version of the Poisson log likelihood function. This replacement allows for very simple proofs of the error bounds. Numerical experiments are performed showing the practical use of the technique in signal and image reconstruction from compressed measurements under Poisson noise.