The identification of a low-dimensional coordinate representation is difficult for high-dimensional, stochastic systems particularly when experience with and intuition for the problem are lacking. In the diffusion map approach, data points generated by simulation are treated as nodes on a weighted graph with edge weights defined by a matrix of pairwise affinities between points (a "kernel"). Appropriate kernel normalization produces a Markov matrix the eigenvectors and eigenvalues of which provide meaningful information on the dataset geometry. We use the first few eigenvectors to provide a coarse grained low dimensional representation for long time evolution of high-dimensional, stochastic systems. This computational technique thereby automates the discovery of good reaction coordinates by performing eigen-processing of detailed data from simulation bursts.

Here we discuss the interpretation of the diffusion map as a random walk on a weighted graph constructed from simulation data and how such an approach incorporates the local dataset geometry and density at each point to build a global picture of the dataset. We illustrate, for a model system, the convergence of diffusion map eigenvectors to the eigenfunctions of differential operators. The use of different diffusion kernels (different Markov chain normalizations) is shown to lead to different limiting differential operators.

We describe lifting and restriction operators for translating between physical variables (of the original system) and diffusion map variables and how these operators facilitate accelerated exploration of the configuration space. We also present "on-demand" estimation of an “effective potential” using this approach for an example using a Molecular Dynamics (MD) simulator. We compare the variables obtained through data-mining direct simulation results, using the diffusion map approach, with an experience-based “intelligent” selection for the coarse variables.

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