Principal sub-manifolds and beyond by Zhigang Yao
Abstract: While classical statistics has dealt with observations which are real numbers or elements of a real vector space, nowadays many statistical problems of high interest in the sciences deal with the analysis of data which consist of more complex objects, taking values in spaces which are naturally not (Euclidean) vector spaces but which still feature some geometric structure. I will discuss the problem of finding principal components to the multivariate datasets, that lie on an embedded nonlinear Riemannian manifold within the higher-dimensional space. The aim is to extend the geometric interpretation of PCA, while being able to capture the non-geodesic form of variation in the data. I will introduce the concept of a principal sub-manifold, a manifold passing through the center of the data, and at any point on the manifold extending in the direction of highest variation in the space spanned by the eigenvectors of the local tangent space PCA. We show the principal sub-manifold yields the usual principal components in Euclidean space. We illustrate how to find, use and interpret the principal sub-manifold, by which a principal boundary can be further defined for data sets on manifolds.
Bio: Zhigang Yao is an Associate Professor in the Department of Statistics and Data Science at the National University of Singapore (NUS). His current research is focused on the interface between statistics and geometry especially on the manifold fitting problem. Currently he is a member of the Center of Mathematical Sciences and Applications at Harvard University. He also holds a courtesy joint appointment with the Department of Mathematics at NUS. He is a Faculty Affiliate of the Institute of Data Science (IDS) at NUS. He has held several visiting positions including Visiting Professorship at EPFL.
He received his Ph.D. in Statistics from University of Pittsburgh in 2011. His thesis advisors are Bill Eddy at Carnegie Mellon and Leon Gleser at University of Pittsburgh. He has been an Assistant Professor at NUS from 2014-2020. Before joining NUS, he has been working with Victor Panaretos as a post-doc researcher at EPFL from 2011-2014.