Nonlinear eigenproblems for high-dimensional data analysis

线性特征值问题是统计学、机器学习、图像处理等领域中大量数据分析方法的基础。非线性特征值问题在非线性泛函分析中已经从理论的角度研究了几十年。然而，它们还没有成为数据分析或其他领域的常用工具，尽管非线性特征值问题提供了额外的建模能力，可以用来加强特征向量的更强属性，如稀疏性或对离群值的鲁棒性，这在线性特征值问题中是不可能的。其原因似乎是缺乏算法所产生的非凸的，往往是非光滑的非线性特征向量的计算优化问题。因此，本研究建议的目的是开发一个通用的算法工具箱来计算非线性特征向量。由于今天的问题，在数据分析的特点是高维的特征空间和大量的样本，我们特别强调的是，可以科普这种情况下，有效的方法的发展。在项目的第二阶段，我们的目标是机器学习，统计和图像处理中的非线性特征问题的具体应用。

Linear eigenproblems are the basis of a large number of methods for data analysis in statistics,machine learning, image processing and many other fields. Nonlinear eigenproblems have been studied from a theoretical point of view in nonlinear functional analysis for decades. Nevertheless, they have not become a common tool in data analysis or other domains, despite the fact that nonlinear eigenproblems provide additional modeling power which can be used to enforce stronger properties of the eigenvectors like sparsity or robustness against outliers which is not possible in the linear eigenproblem. The reason seems to be the lack of algorithms for the arising nonconvex and often nonsmooth optimization problems in the computation of nonlinear eigenvectors. Thus, the purpose of this research proposal is to develop a general toolbox of algorithms to compute nonlinear eigenvectors. As today’s problems in data analysis are characterized by both high dimensionality of the feature space and a large number of samples, we place particular emphasis on the development of efficient methods which can cope with such cases. In the second phase of the project we target specific applications of nonlinear eigenproblems in machine learning, statistics and image processing.