A Data Driven Orthonormal Parameterization of the Generalized Entropy Maximization Problem

During the past years Lindquist and coworkers have formulated and studied the so called generalized maximum entropy problem. It generalizes the maximum entropy problem and thus includes problem like the Carathéodory extension problem and the Nevanlinna-Pick interpolation problem. A central ingredient in the theory is the dual (in mathematical programming sense) convex optimization problem. This paper studies a re-parameterization with orthonormal basis functions of that optimization problem. The basis functions can be chosen independent of the original problem; for instance they can be chosen to be data driven. In numerical examples it is shown that a sensible choice of basis functions can improve the numerical conditioning of the optimization problem.