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 into gradient, harmonic, and curl, known as the Helmholtz-Hodge Decomposition, provides a basis for all vector fields on a manifold. We present an estimator for $\Delta_1$, and based on it we develop a variety of applications. Among them, visualization of the principal harmonic, gradient or curl flows on a manifold, smoothing and semi-supervised learning of vector fields, 1-Laplacian regularization. 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, Samson Koelle, Hanyu Zhang and Ioannis Kevrekidis