Solution of linear ill-posed problems using flexible dictionaries

In the present paper we consider application of flexible, overcomplete dictionaries to solution of general ill-posed linear inverse problems. Construction of an adaptive optimal solution for problems of this sort usually relies either on a singular value decomposition or representation of the solution via some orthonormal basis. The shortcoming of both approaches lies in the fact that, in many situations, neither the eigenbasis of the linear operator nor a standard orthonormal basis constitutes an appropriate collection of functions for sparse representation of f. In the context of regression problems, there have been enormous amount of effort to recover an unknown function using a flexible, overcomplete dictionary. One of the most popular methods, Lasso and its versions, is based on minimizing empirical likelihood and, unfortunately, requires stringent assumptions on the dictionary, the, so called, compatibility conditions. While these conditions may be satisfied for the functions in the original dictionary, they usually do not hold for their images due to contraction imposed by the linear operator. In the paper, we bypass this difficulty by a novel approach which is based on inverting each of the dictionary functions and matching the resulting expansion to the true function rather than minimizing the empirical likelihood, thus, avoiding unrealistic assumptions on the dictionary. We show how the suggested methodology can be extended to the problem of estimation of a mixing density in a continuous mixture. We also suggest the solution which utilizes structured and unstructured random dictionaries, the technique that have not been applied so far to the solution of ill-posed linear inverse problems. We put a solid theoretical foundation under the suggested methodology and study its performance via simulations that confirm good computational properties of the method.