支持函数机及其应用

函数型数据是一类重要的数据类型，广泛存在于医学、气象学和生物信息学等领域。当机器学习中的输入为函数型数据时，学习问题就变成了函数型数据的学习问题。当前学习机大多仅适用于向量型数据的学习问题，因此为函数型数据的学习问题构建合适的学习机，便成为亟待研究的问题。本项目主要以最优化理论和方法为指导，研究函数型数据的分类问题、回归问题和顺序回归问题，建立相应的新的支持函数机的模型和算法。这里的函数型数据的形式包括单个指标的和多个指标的，以及更广泛的函数型数据与向量型数据的混合。特别关注学习机中的正则项技术和核技术引入，以提高模型的泛化能力，且提高模型的稀疏性和可解释性。同时深入研究相应的优化理论并设计相应的快速有效地求解算法。此外，还要考虑如何将支持函数机应用于某种疾病预测中。本项目的研究内容是最优化方法、函数型数据分析和机器学习这三个研究领域的交叉，具有一定的创新性和挑战性。

Functional data is an important type of data, which widely exists in practical fields such as medicine, meteorology, and bioinformatics. When the input data in machine learning are functional data, machine learning becomes a learning problem of functional. At present, most learning machines are only suitable for learning problems of vectorial data. Therefore, building a learning machine suitable for the functional data has become an urgent problem. Using optimization theory and methods, this project studies the classification problem, regression problem and ordinal regression problem of the functional data, and establishes new support function machines, including their models and algorithms. Here the data allow being functional data with both single feature and multiple features, as well as the extensive mixture of functional data and usual vectorial data. Special attention is paid to the regularization technique and kernel technique to improve the generalization ability, and enhance the sparsity and the interpretability of the learning machine. The corresponding optimization theory and effective algorithms are also considered. In addition, the support function machines are also applied to some disease prediction. The research of this project is the intersection of the three research fields of optimization methods, functional data analysis and machine learning. It is innovative and challenging.