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Principal Flows
澳门新葡8455最新网站:2014年11月11日 00:00 点击数:

报告人:姚志刚

报告地点:澳门新葡8455最新网站317室

报告澳门新葡8455最新网站:2014年12月09日星期二14:00-15:00

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报告摘要:

We revisit the problem of extending the notion of principal component analysis (PCA) to multivariate data sets that satisfy non-linear constraints, therefore lying on Riemannian manifolds. Our aim is to determine curves on the manifold that retain their canonical interpretability as principal components, while at the same time being flexible enough to capture non-geodesic forms of variation. We introduce the concept of a principal flow, a curve on the manifold passing through the mean of the data, and with the property that, at any point of the curve, the tangent velocity vector attempts to fit the first eigenvector of a tangent space PCA locally at that same point, subject to a smoothness constraint. That is, a particle owing along the principal flow attempts to move along a path of maximal variation of the data, up to smoothness constraints. The rigorous definition of a principal flow is given by means of a Lagrangian variational problem, and its solution is reduced to an ODE problem via the Euler-Lagrange method. Conditions for existence and uniqueness are provided, and an algorithm is outlined for the numerical solution of the problem. Higher order principal flows are also defined. It is shown that principal flows can yield the usual principal components on a Euclidean space. By means of examples, it is illustrated that the principal flow is able to capture patterns of variation that can escape other manifold PCA methods. This is a joint work with Victor Panaretos and Tung Pham

主讲人概况:

Zhigang Yao is an assistant professor in the Department of Statistics and Applied Probability at the National University of Singapore (NUS). 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. Before joining NUS, he has been working with Victor Panaretos as a post-doc researcher at the Swiss Federal Institute of Technology (EPFL) from 2011-2014.

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