Mathematical Sciences: Topics in Projection Pursuit, Neural Networks and Pattern Recognition

本课程将研究统计学、数学和计算科学交界处的问题，重点是回归和密度函数的傅里叶近似和岭估计，以及在不同的背景下，用于局部平滑的过滤狄里克莱特分布。特别地，将研究多傅立叶展开中的频率权重和单隐层神经网络中的连接权重的偏好顺序。文中还将详细讨论特征提取和投影寻踪等应用。这项研究将有助于从机器人学、医学和人工智能等大数据集中进行自动学习。这项工作的一个方面是开发复杂的数学技术，用于统计和电气工程中的问题。这项工作的第二个方面将开发统计方法，以便直接利用专家的知识，而不限制专家必须表达这一信息的形式。

Problems in the interface of statistics, mathematics and computational science will be studied, focusing on Fourier approximation and ridge estimation of regression and density functions and, in a separate context, on filtered Dirichlet distributions for local smoothing. In particular, preference orders will be studied for the frequency weights in multiple Fourier expansions and for the connection weights in single hidden-layer neural networks. Properties will be derived and applications such as feature extraction and projection pursuit will be considered in detail. This research will aid in automated learning from large data sets such as those arising in robotics, medicine and artificial intelligence. One aspect of the work will develop sophisticated mathematical techniques for application to problems in statistics and electrical engineering. A second aspect of the work will develop statistical methods to utilize an expert's knowledge directly without restricting the form in which the expert must express this information.