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

最近凸优化的革命已经反映在统计机器学习中，它也能够改变其他统计学科的格局，特别是函数响应的非参数拟合。在这种思想框架下，该提案旨在开发和研究功能数据分析的计算机密集型方法，特别是具有L1正则化风格的各种提案；主题包括功能分位数估计、不规则观测时间序列的频谱分析、功能响应模式的识别、非标准响应的分位数回归、半参数密度估计等。假设优化主题不仅在数值实现中发挥重要作用，而且在理论中发挥重要作用；其中一个提议的领域涉及由凸优化定义的统计过程的共轭对偶公式的作用。探究的一个特定部分，超越了功能拟合，将致力于将复杂对象拟合到数据的概念问题。所提出的研究的预期结果将是工作的统计方法，具有更广泛的范围，更高的灵敏度和增强的稳健性，适用于与统计函数拟合相关的各个领域。

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.