This talk will extend Manifold Learning 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, we ask how popular Manifold Learning tools and their applications can be recreated in the space of  vector fields and flows on a manifold. For this, we need to transport= advanced differential geometric and topological concepts into a data-driven framework.

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. The algorithms are illustrated on a variety of real data sets.

Joint work with  Yu-Chia Chen, Samson Koelle, Vlad Murad, Weicheng Wu, Hanyu Zhang and Ioannis Kevrekidis