Diffusion maps tailored to arbitrary non-degenerate Itô processes

Abstract

We present two generalizations of the popular diffusion maps algorithm. The first generalization replaces the drift term in diffusion maps, which is the gradient of the sampling density, with the gradient of an arbitrary density of interest which is known up to a normalization constant. The second generalization allows for a diffusion map type approximation of the forward and backward generators of general Itô diffusions with given drift and diffusion coefficients. We use the local kernels introduced by Berry and Sauer, but allow for arbitrary sampling densities. We provide numerical illustrations to demonstrate that this opens up many new applications for diffusion maps as a tool to organize point cloud data, including biased or corrupted samples, dimension reduction for dynamical systems, detection of almost invariant regions in flow fields, and importance sampling.

Local and global perspectives on diffusion maps in the analysis of molecular systems

“…where ∆ is the Laplace-Beltrami operator on M and ∇ is the gradient operator on M. Note that in the special case α = 1/2, and for the choice π = Z −1 e −βV (Boltzmann-Gibbs), the approximated operator corresponds to the generator of overdamped Langevin dynamics (4), such that…”

“…We use 10 4 points obtained by sub-sampling of the trajectory for the diffusion maps, which is chosen with kernel (6) and ε = 0.1. We compute the first two dominant eigenvectors and define the metastable subsets A and B using the first dominant eigenvector as described above 4 . In Figure 7(a) we see the sampling and the chosen sets.…”

“…For this purpose, the target measure diffusion map was recently introduced [19]. The main advantage is that it allows construction of an approximation of L even if the data points are distributed with respect to some distribution μ such that supp π ⊂ supp μ.…”

“…The spectral decomposition of the diffusion map matrix thus yields an approximation of the continuous spectral problem on the point-cloud [42] and leads to natural collective variables. Since the first appearance of diffusion maps [17], several improvements have been proposed including local scaling [50], variable bandwidth kernels [7] and target measure maps (TMDmap) [4].The latter scheme extends diffusion maps on point-clouds obtained from a surrogate distribution, ideally one that is easier to sample from. Based on the idea of importance sampling, it can be used on biased trajectories, and improves the accuracy and application of diffusion maps in high dimensions [4].…”

“…Since the first appearance of diffusion maps [17], several improvements have been proposed including local scaling [50], variable bandwidth kernels [7] and target measure maps (TMDmap) [4].The latter scheme extends diffusion maps on point-clouds obtained from a surrogate distribution, ideally one that is easier to sample from. Based on the idea of importance sampling, it can be used on biased trajectories, and improves the accuracy and application of diffusion maps in high dimensions [4]. Diffusion maps can be combined with the variational approach to conformation dynamics in order to improve the approximation of the eigenfunctions and provide eigenvalues that relate directly to physical relaxation timescales [10].…”