Overview
This is an ongoing collaboration with Cecilia Clementi and various people at her lab at Rice. See our recent publications:- M. A. Rohrdanz, W. Zheng, M. Maggioni, C. Clementi, Determination of reaction coordinates via locally scaled diffusion map. J. Chem. Phys., 134 2011: 124116
- W. Zheng, M. A. Rohrdanz, M. Maggioni, C. Clementi, Polymer reversal rate calculated via locally scaled diffusion map. J. Chem. Phys., 134 2011: 144108
See also the snippet at the Institute of Pure and Applied Mathematics, where this research initiated, here
The main ideas are that data from molecular dynamics simulations, e.g. in the form of the coordinates of the atoms in a molecule as a function of time, lie on or near an intrinsically-low-dimensional set in the high-dimensional state space of the molecule, and geometric properties of such sets provide important information about the dynamics, or about how to build low-dimensional representations of such dynamics. We apply recent work on estimation of the intrinsic dimension of data sets in high-dimensions to such data, validating the hypothesis that indeed the set of configurations of the molecule does indeed lie on an intrinsically low-dimensional set (at least, in the examples considered), and then use this information, together with a notion of local scale (roughly defined as the largest scale where the data is well-approximated by a low-dimensional linear subspace), to introduce a variation of diffusion maps, that leads to a set of nonlinear coordinates in state space onto which we may project the dynamics and construct a low-dimensional diffusion process that well-approximates the large-time behavior of the molecular dynamics simulation.
See this nugget and the papers above for more information.
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