Academic

Manifold learning algorithms can recover the underlying low-dimensional parametrization of high-dimensional point clouds. This talk will extend this paradigm in two directions. First, we ask if it is possible to explain a data manifold by new coordinates chosen from a set of scientifically meaningful functions where quantitative prior knowledge is abundant. Second, we will show how some popular manifold learning tools and applications can be recreated in the space of vector fields and flows on a manifold. We further present an estimator for the first order Laplacian of a manifold . Under the topological data analysis, we describe the first order analogue of the spectral clustering, which amounts to prime manifold decomposition. A new algorithm for finding the shortest independent loops follows from this decomposition. We illustrate this algorithm using a variety of real data sets.

Joint work with Yu-Chia Chen, Samson Koelle, Hanyu Zhang and Ioannis Kevrekidis.