Low-Rank Matrix Completion with Poisson Observations via Nuclear Norm and Total Variation Constraints (Journal of Computational Mathematics, 2024)
In this paper, we study the low-rank matrix completion problem with Poisson observations, where only partial entries are available and the observations are in the presenceof Poisson noise. We propose a novel model composed of the Kullback-Leibler (KL) divergence by using the maximum likelihood estimation of Poisson noise, and total variation(TV) and nuclear norm constraints. Here the nuclear norm and TV constraints are utilized to explore the approximate low-rankness and piecewise smoothness of the underlyingmatrix, respectively. The advantage of these two constraints in the proposed model is thatthe low-rankness and piecewise smoothness of the underlying matrix can be exploited simultaneously, and they can be regularized for many real-world image data. An upper errorbound of the estimator of the proposed model is established with high probability, which isnot larger than that of only TV or nuclear norm constraint. To the best of our knowledge,this is the first work to utilize both low-rank and TV constraints with theoretical errorbounds for matrix completion under Poisson observations. Extensive numerical exampleson both synthetic data and real-world images are reported to corroborate the superiorityof the proposed approach.
Keywords: Low-rank matrix completion; Nuclear norm; Total variation; Poisson observations.