Nonlinear eigenproblems for high-dimensional data analysis

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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.