Dictionary Learning for the non-linear approximation of spherical functions

在之前的项目中，开发了一种名为ROFMP的算法，它能够正则化逆问题（特别是在球面上）。从初步选择的所谓试函数字典中，迭代地选择这些函数，它们共同构成了所考虑的反问题的一种“最佳基”。出于这个原因，ROFMP是贪婪算法的一个例子。由于ROFMP，可以将几种类型的试验函数的优点结合起来。例如，这使得处理极不不规则分布的数据网格和构建解决方案的多尺度分析成为可能。此外，与样条法相比，ROFMP产生了至少同样好的解决方案，然而，本质上较少的试验函数。除其他外，还应用于处理重力场的卫星数据，这是地球的重要参考。到目前为止，可用的试验函数字典是根据经验先验地选择的。在后续项目中，应开发一种方法，使词典的选择自动化。对于主要用于图像处理的贪心算法，也有相关的技术，在关键词字典学习下进行了总结。然而，这些方法中的绝大多数都是针对离散函数设计的。因此，字典仅仅被表示为一个矩阵，方法是基于数值线性代数的技术。这种方法对于设想的应用程序没有意义，因为使用的试验函数通常具有物理解释。因此，在解决方案的表示中维护这些函数使应用科学家能够适当地解释所获得的结果。此外，所要解决的问题用典型的分析方程如积分方程来表示。因此，离散化从一开始就会给计算带来不准确性。因此，该算法需要在概念上进行新的工作。它应包括两个级别，考虑到可用的试验函数可以细分为不同的类型。算法的上层决定哪些类型对所考虑的问题有用。在较低的层次上，开发了子算法，在每个单一类型中进行最优选择。这可以通过特定类型的参数来控制。为了实现，优化算法需要与基于经验的启发式相结合。该项目的一个目标是为典型的应用场景制作一个“最优”的初始字典，例如对重力场变化（由于气候）的分析。从相应的应用中，科学家们可以很容易地使用这本词典。

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.