Non-negative differentially constrained entropy-like regularization

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Abstract
Classical maximum entropy is quite popular, not only because it is based on appropriate physical and phenomenological concepts, but also because it yields, among other things, a natural way in which to invoke, or even force, a positivity constraint. However, though it is known to converge with reasonable rates of convergence and to exhibit good stability properties, it is, from a practical point of view, flawed, because of its inherent shrinkage properties. This has been well documented along with the fact that it can, when the circumstances are appropriate, exhibit super-resolution behaviour. Thus, there is a clear need to examine whether the maximum entropy functional can be modified in a way which, on the one hand, removes the shrinkage difficulty and, on the other hand, guarantees good asymptotics for the oversmoothed solution as well as reasonable rates of convergence and appropriate stability. This is the goal of the present paper. In particular, it is shown how such results are obtained by simply replacing the function u in the classical entropy regularizor by a differential operator , which may have a non-trivial null space. Such regularizors will be called non-negative differentially constrained entropy-like regularizors. The utility of such regularizors is examined not only in terms of the standard stability, convergence and rates of convergence, but also in terms of the important practical measures of asymptotics and shrinkage. In particular, it is shown that the unwanted shrinkage does not, in general, hold for such entropy-like regularizors, the choice of which can be more specifically adapted to the problem context of the regularization than its classical counterpart. Some practical aspects of these entropy-like regularizors are briefly discussed in the introduction. In particular, because such regularizors automatically guarantee the positivity of , the regularized approximations thereby generated will retain the monotonicity or convexity of the underlying solution u if is taken to be or , respectively.