Abstract:

Motivated by problems in structural biology, specifically cryo-electron 
microscopy, we introduce vector diffusion maps (VDM), a new mathematical 
framework for organizing and analyzing high dimensional data sets, 
2D images and 3D shapes. VDM is a mathematical and algorithmic
generalization of diffusion maps and other non-linear dimensionality
reduction methods, such as LLE, ISOMAP and Laplacian eigenmaps. While
existing methods are either directly or indirectly related to the heat
kernel for functions over the data, VDM is based on the heat kernel for
vector fields. VDM provides tools for organizing complex data sets,
embedding them in a low dimensional space and interpolating and regressing
vector fields over the data. In particular, it equips the data with a
metric, which we refer to as the vector diffusion distance. In the manifold
learning setup, where the data set is distributed on a low dimensional
manifold M^d embedded in R^p, we prove the relationship between VDM and the
connection-Laplacian operator for vector fields over the manifold.
Applications to structural biology (cryo-electron microscopy and NMR
spectroscopy), computer vision and shape space analysis will be discussed.

(Joint work with Hau-tieng Wu.)

Short Bio:

Amit Singer is an Associate Professor of Mathematics and a member of the
Executive Committee of the Program in Applied and Computational Mathematics
(PACM) at Princeton University. He joined Princeton as an Assistant
Professor in 2008. From 2005 to 2008 he was a Gibbs Assistant Professor in
Applied Mathematics at the Department of Mathematics, Yale University.
Singer received the BSc degree in Physics and Mathematics and the PhD degree
in Applied Mathematics from Tel Aviv University (Israel), in 1997 and 2005,
respectively. He served in the Israeli Defense Forces during 1997-2003. He
was awarded the Presidential Early Career Award for Scientists and Engineers
(2010), the Alfred P. Sloan Research Fellowship (2010) and the Haim Nessyahu
Prize for Best PhD in Mathematics in Israel (2007). His current research in
applied mathematics focuses on problems of massive data analysis and
structural biology.


Refreshments served from 14:15 on,
 	Lecture starts at 14:30