Fast Algorithm for Low-rank matrix recovery in Poisson noise

ABSTRACT This paper describes a new algorithm for recovering low-rank matrices from their linear measurements contaminated with Poisson noise: the Poisson noise Maximum Likelihood Singular V alue thresholding (PMLSV) algorithm. We propose a convex optimization formulation with a cost function consisting of the sum of a likelihood function and a regularization function which is proportional to the nuclear norm of the matrix. Instead of solving the optimization problem directly by semi-definite program (SDP), we derive an iterative singular value thresholding algorithm by expanding the likelihood function. We demonstrate the good performance of the proposed algorithm on recovery of solar flare images with Poisson noise: the algorithm is more efficient than solving SDP using the interior-point algorithm and it generates a good approximate solution compared to that solved from SDP . Index Terms-- low-rank matrix recovery, nuclear norm, singular value thresholding, solar flare images 1. INTRODUCTION Recovery of a matrixM from its linear measurements (or linear projections) contaminated with Poisson noise arises from various important applications such as optical imaging, nuclear medicine and X-ray imaging [1].