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Spectral estimation and inverse initial boundary value problems
A theoretical framework for the regularization of Poisson likelihood estimation problems
1.  Department of Mathematical Sciences, University of Montana, Missoula, Montana 59812, United States 
$R_\alpha(A,z)= arg\min_{u\geq 0} \{T_0(Au;z)+\alpha J(u)\},$
where $T_0(Au;z)$ is the negativelog of the Poisson likelihood functional, and $\alpha>0$ and $J$ are the regularization parameter and functional, respectively. Our goal in this paper is to determine general conditions which guarantee that $R_\alpha$ defines a regularization scheme for $z=Au+\gamma$. Determining the appropriate definition for regularization scheme in this context is important: not only will it serve to unify previous theoretical arguments in this direction, it will provide a framework for future theoretical analyses. To illustrate the latter, we end the paper with an application of the general framework to a case in which an analysis has not been done.
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