Optimization theory and algorithms in functional and object-oriented data analysis: from quantitative to qualitative aspects

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The recent revolution in convex optimization, already reflected in statistical machine learning, is capable of changing also the landscape of other statistical disciplines, in particular nonparametric fitting of functional responses. The proposal aims, in this frame of mind, at developing and investigating computer-intensive methods of functional data analysis, especially various proposals with L1 regularization flavor; the topics comprise, among others, functional quantile estimation, spectral analysis of irregularly observed time series, recognition of patterns in functional responses, quantile regression with nonstandard responses, semiparametric density estimation, and others. The optimization theme is presumed to play a significant role not only in numerical implementations, but also in the theory; one of the proposed areas concerns the role of conjugate dual formulation of the statistical procedures defined by convex optimization. A specific part of inquiry, transcending from functional fitting, will be devoted to the conceptual problems of fitting complex objects to data. The expected outcome of the proposed research will be working statistical methods, with broader scope, greater sensitivity, and increased robustness, applicable in various areas related to statistical functional fitting.