Dictionary Learning for the non-linear approximation of spherical functions

球函数非线性逼近的字典学习

• 批准号: 169129297
• 负责人: Professor Dr. Volker Michel
• 金额: --
• 依托单位: Arbeitsgruppe Geomathematik
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2010
• 资助国家: 德国
• 起止时间: 2009-12-31 至 2021-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/169129297?language=en
• 关键词: Dictionary; Learning; non; linear; approximation

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.

在上一个项目中，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）。从一个初步选择的所谓字典的试验功能，这些功能被反复选择，共同构成一种“最佳基础”的考虑逆问题。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用功能的优点。例如，这使得能够处理非常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，例如，与样条方法相比，ROFMP产生至少同样好的解，然而，基本上较少的试验函数。应用包括处理卫星重力场数据，这是地球的重要参考。到目前为止，可用试函数的字典是根据经验先验选择的。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相关贪婪算法，存在这样的技术，这些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是为离散函数设计的。因此，字典仅仅被表示为矩阵，并且方法基于数值线性代数的技术。这样的方法对于所设想的应用没有意义，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，所要解决的问题是由典型的分析方程，如积分方程。因此，离散化将从一开始就在计算中加入不准确性，因此，该算法需要在概念上进行新的工作。它应包括两个层次，考虑到现有的审判职能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，子算法的开发，使每一个单一的类型之间的最佳选择。这可以通过特定类型的参数进行控制。在实施过程中，应将优化算法与基于经验的算法相结合。该项目的一个目标是为一个典型的应用场景，如分析重力场变化（由于气候），提供一个“最佳”的初始字典。该词典可以很容易地被科学家从相应的应用程序中使用。