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

最近的凸优化革命，已经反映在统计机器学习中，也能够改变其他统计学科的景观，特别是函数响应的非参数拟合。该提案的目的是，在这种心态下，在开发和研究功能数据分析的计算机密集型方法，特别是各种建议与L1正则化风味;主题包括，除其他外，功能分位数估计，不规则观测时间序列的谱分析，识别功能响应的模式，分位数回归与非标准响应，半参数密度估计等。最优化的主题被认为不仅在数值实现中，而且在理论中发挥了重要作用;提出的领域之一涉及由凸优化定义的统计过程的共轭对偶公式的作用。调查的一个特定部分，超越功能拟合，将致力于拟合复杂对象的数据的概念问题。拟议研究的预期成果将是适用于与统计函数拟合相关的各个领域的范围更广、灵敏度更高、鲁棒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.