Dictionary Learning for the non-linear approximation of spherical functions

在先前的项目中，我们开发了一种名为ROFMP的算法，该算法能够正则化反问题(特别是在球面上)。从初步选定的所谓试函数词典中，迭代地选择那些函数，这些函数共同构成所考虑的反问题的一种“最佳基”。因此，ROFMP是贪婪算法的一个例子。由于ROFMP，可以结合几种类型的试函数的优点。例如，这使得能够处理极不规则分布的数据网格和构建解决方案的多尺度分析。此外，与例如样条法相比，ROFMP得到了至少同样好的解，然而，试函数基本上更少。除其他应用外，重力场卫星数据的处理是地球的一个重要参考。到目前为止，可用的试函数词典是根据经验先验选择的。在后续项目中，应开发一种自动选择词典的方法。对于主要用于图像处理的相关贪婪算法，也存在这样的技术，在关键字词典学习一节中总结了这些技术。然而，这些方法中的绝大多数都是为离散化函数设计的。因此，字典仅仅被表示为一个矩阵，并且这些方法基于数值线性代数技术，这种方法对于设想的应用是没有意义的，因为所使用的试函数通常具有物理解释。因此，在解的表示中保持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，要解决的问题由积分方程组等典型的分析方程来表示。因此，离散化将从一开始就给计算带来不准确性。因此，该算法需要从概念上进行新的工作。它应由两个级别组成，这两个级别考虑到可用的试验功能可以细分为不同的类型。算法的上层决定了哪些类型对所考虑的问题是有用的。在较低层，开发了子算法，在每种类型中进行最优选择。这可以通过特定于类型的参数进行控制。在实现上，优化算法应与基于经验的启发式算法相结合。该项目的一个目标是为一个典型的应用场景，例如分析引力场变化(由于气候)，提供一个“最佳”的初始词典。这本词典很容易被科学家从相应的应用程序中使用。

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.