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

凸优化的最新革命已经反映在统计机器学习中，也能够改变其他统计学科的格局，特别是功能反应的非参数拟合。在这种心态下，该提案旨在开发和研究计算机密集的函数数据分析方法，特别是具有L1正则化特征的各种方案；这些主题包括函数分位数估计、不规则观察的时间序列的谱分析、功能响应模式的识别、带有非标准响应的分位数回归、半参数密度估计等。优化主题被认为不仅在数值实现中，而且在理论上都发挥着重要的作用；其中一个拟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。研究的一个特定部分，超越了函数拟合，将致力于将复杂对象与数据拟合的概念性问题。拟议研究的预期成果将是适用于与统计函数拟合有关的各个领域的适用范围更广、灵敏度更高、稳健性更强的工作统计方法。

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.