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

在统计，数学和计算科学的接口问题将被研究，集中在傅里叶近似和脊估计的回归和密度函数，并在一个单独的背景下，对局部平滑过滤狄利克雷分布。特别是，优先顺序将研究在多次傅立叶展开中的频率权值和在单个隐藏层神经网络中的连接权值。我们将推导出属性，并详细讨论特征提取和投影追踪等应用。这项研究将有助于从机器人、医学和人工智能等领域的大型数据集中进行自动学习。这项工作的一个方面将发展复杂的数学技术，用于解决统计学和电气工程中的问题。这项工作的第二个方面将发展统计方法，直接利用专家的知识，而不限制专家必须表达这些信息的形式。

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.