In recent years, manifold learning has become increasingly popular as a tool for performing non-linear dimensionality reduction and for discovering collective variables. I will present a set of advanced manifold learning methods, that all aim to uncover and preserve the geometric properties of the data.
Among these will be a method to estimate vector fields on a manifold, estimation of the kernel width and intrinsic dimension, a relaxation method to remove distortions induced by the embedding algorithms. These methods all build on on the Diffusion maps framework, and exploit the relationship between the Laplace-Beltrami operator and the Riemannian metric on a manifold.
Finally, I will show some preliminary results of finding low dimensinal embeddings of molecular fingerprints.
Joint work with Dominique Perrault-Joncas, James McQueen, Jacob VanderPlas, Zhongyue Zhang, Grace Telford, Oles Isayev, Alexandre Tkatchenko, Stefan Chmiela