Dictionary Learning for the non-linear approximation of spherical functions

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In the previous project, an algorithm named ROFMP was developed, which is able to regularize inverse problems (in particular, on the sphere). From a preliminarily chosen so-called dictionary of trial functions, those functions are chosen iteratively which constitute together a kind of a "best basis" for the considered inverse problem. For this reason, the ROFMP is an example of a greedy algorithm. Due to the ROFMP, the advantages of several types of trial functions can be combined. This enables, for example, the handling of extremely irregularly distributed data grids and a construction of a multiscale analysis of the solution. Furthermore, in comparison to, for instance, a spline method, the ROFMP yields an at least equally good solution with, however, essentially less trial functions. Applications occur, amongst others, for the treatment of satellite data of the gravitational field, which is an important reference for the Earth.So far, the dictionary of available trial functions is chosen a priori based on experience. In the follow-on project, a method shall be developed which automatizes the choice of the dictionary. For related greedy algorithms which are primarily used for image processing there exist such techniques, which are summarized under the key word dictionary learning. However, the vast majority of these methods is designed for discretized functions. As a consequence, the dictionary is merely represented as a matrix, and the methods are based on techniques of Numerical Linear Algebra.Such approaches do not make sense for the envisioned applications since the used trial functions usually have a physical interpretation. Therefore, the maintenance of these functions in the representation of the solution enables the applied scientists to interpret the obtained result appropriately. Furthermore, the problems to be solved are represented by typical equations of analysis such as integral equations. Hence, a discretization would install an inaccuracy into the calculations from the very beginning.Thus, the algorithm requires conceptually new work. It shall consist of two levels which take into account that the available trial functions can be subdivided into separate types. The upper level of the algorithm decides which types are useful for the considered problem. At the lower level, subalgorithms are developed, which make an optimal selection among each single type. This can be controlled via type-specific parameters. For the implementation, optimization algorithms shall be combined with experience-based heuristics. One aim of the project is to make an "optimal" initial dictionary for a typical application scenario, such as the analysis of gravitational field variations (due to climate), available. This dictionary can easily be used by scientists from the corresponding applications.