  The study of rare events in molecular and atomic systems such as conformal
changes and cluster rearrangements has been one of the most important research
themes in chemical physics. Key challenges are associated with long waiting
times rendering molecular simulations inefficient, high dimensionality impeding
the use of PDE-based approaches, and the complexity or breadth of transition
processes limiting the predictive power of asymptotic methods. Diffusion maps
are promising algorithms to avoid or mitigate all these issues. We adapt the
diffusion map with Mahalanobis kernel proposed by Singer and Coifman (2008) for
the SDE describing molecular dynamics in collective variables in which the
diffusion matrix is position-dependent and, unlike the case considered by
Singer and Coifman, is not associated with a diffeomorphism. We offer an
elementary proof showing that one can approximate the generator for this SDE
discretized to a point cloud via the Mahalanobis diffusion map. We use it to
calculate the committor functions in collective variables for two benchmark
systems: alanine dipeptide, and Lennard-Jones-7 in 2D. For validating our
committor results, we compare our committor functions to the finite-difference
solution or by conducting a "committor analysis" as used by molecular dynamics
practitioners. We contrast the outputs of the Mahalanobis diffusion map with
those of the standard diffusion map with isotropic kernel and show that the
former gives significantly more accurate estimates for the committors than the
latter.
