讲座题目：Principal Sub-manifolds

主讲人：新加坡国立大学 姚志刚博士 

讲座时间: 2017年12月11日（星期一）3:00-4:00pm 

讲座地点：浙江工商大学综合楼601会议室

主讲人简介：姚志刚博士，2011年获美国匹兹堡大学统计学博士学位，此后于2011-2014年在瑞士联邦理工学院(EPFL)从事博士研究员。现为新加坡国立大学统计与应统概率系任教。主要从事复杂数据（包括反问题,高纬数据，流型数据)的研究。  先后主持美国国家科学基金，新加坡教育部(MOE)和新加坡国立大学(NUS)科学基金多项。  

摘要：

     We revisit the problem of finding principal  components to the multivariate datasets, that lie on an embedded nonlinear  Riemannian manifold within the higher-dimensional space. Our aim is to extend  the geometric interpretation of PCA, while being able to capture the  nongeodesic form of variation in the data. We introduce the concept of a  principal sub-manifold, a manifold passing through the center of the data, and  at any point of the manifold, it moves in the direction of the highest curvature  in the space spanned by the eigenvectors of the local tangent space PCA.  Compared to the recent work in the case where the sub-manifold is of dimension  one (Panaretos et al. 2014), essentially a curve lying on the manifold  attempting to capture the one-dimensional variation, the current setting is much  more general. The principal sub-manifold is therefore an extension of the  principal flow, accommodating to capture the higher dimensional variation in the  data. We show the principal sub-manifold yields the usual principal components  in Euclidean space. By means of examples, we illustrate that how to find, use  and interpret principal sub-manifold with an extension of using it in shape  analysis. (This is a joint work with Tung Pham)