  Diffusion maps approximate the generator of Langevin dynamics from simulation
data. They afford a means of identifying the slowly-evolving principal modes of
high-dimensional molecular systems. When combined with a biasing mechanism,
diffusion maps can accelerate the sampling of the stationary Boltzmann-Gibbs
distribution. In this work, we contrast the local and global perspectives on
diffusion maps, based on whether or not the data distribution has been fully
explored. In the global setting, we use diffusion maps to identify metastable
sets and to approximate the corresponding committor functions of transitions
between them. We also discuss the use of diffusion maps within the metastable
sets, formalising the locality via the concept of the quasi-stationary
distribution and justifying the convergence of diffusion maps within a local
equilibrium. This perspective allows us to propose an enhanced sampling
algorithm. We demonstrate the practical relevance of these approaches both for
simple models and for molecular dynamics problems (alanine dipeptide and
deca-alanine).
