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, in the case of scientific data where quantitative prior knowledge is abundant, to explain a data manifold by new coordinates, chosen from a set of scientifically meaningful functions. Second, I will show how some popular manifold learning tools and applications can be recreated in the space of vector fields and flows on a manifold. Central to this approach is the order 1-Laplacian of a manifold, $\Delta_1$, whose eigen-decomposition, known as the Helmholtz-Hodge Decomposition, provides a basis for all vector fields on a manifold. We present a consistent estimator for $\Delta_1$, which enables visualization, analysis, smoothing and semi-supervised learning of vector fields on a manifold. In topological data analysis, we describe the 1st-order analogue of spectral clustering, which amounts to prime manifold decomposition. Furthermore, from this decomposition a new algorithm for finding shortest independent loops follows.
Joint work with Yu-Chia Chen, Weicheng Wu, Samson Koelle, Hanyu Zhang and Ioannis Kevrekidis