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A theoretical analysis of L1 regularized Poisson likelihood estimation

A standard variational formulation for solving the linear operator equation Au = z is to compute the function u that minimizes ‖Au − z‖L2(Ω). This is often an ill-posed problem that requires regularization, which is to say that there may exist infinitely many minimizers or that the solution may not be stable with respect to measurement errors. In many cases, however, the measured function z is contaminated by the noise of both Gaussian and Poisson types. If the Poisson noise is mathematically significant but not of high values, then solving the operator equation via the ubiquitous least-squares minimization problem may not produce an appropriate reconstruction, and the Poisson negative log-likelihood estimator may lead to a more accurate result. The Poisson likelihood minimization is also generally an ill-posed problem, and we require a regularization term to impose stability or to pick out which of the infinitely many minimizers is appropriate. Many possible regularizations have been analysed, including Tikhonov and total-variation regularizations, but in this work we perform the theoretical analysis to show that choosing the Poisson minimizer of smallest L1 norm leads to a theoretically well-posed problem.