Sharp error estimates for target measure diffusion maps with
  applications to the committor problem
Shashank Sule Luke Evans Maria Cameron
We obtain asymptotically sharp error estimates for the consistency error of
the Target Measure Diffusion map (TMDmap) (Banisch et al. 2020), a variant of
diffusion maps featuring importance sampling and hence allowing input data
drawn from an arbitrary density. The derived error estimates include the bias
error and the variance error. The resulting convergence rates are consistent
with the approximation theory of graph Laplacians. The key novelty of our
results lies in the explicit quantification of all the prefactors on
leading-order terms. We also prove an error estimate for solutions of Dirichlet
BVPs obtained using TMDmap, showing that the solution error is controlled by
consistency error. We use these results to study an important application of
TMDmap in the analysis of rare events in systems governed by overdamped
Langevin dynamics using the framework of transition path theory (TPT). The
cornerstone ingredient of TPT is the solution of the committor problem, a
boundary value problem for the backward Kolmogorov PDE. Remarkably, we find
that the TMDmap algorithm is particularly suited as a meshless solver to the
committor problem due to the cancellation of several error terms in the
prefactor formula. Furthermore, significant improvements in bias and variance
errors occur when using a quasi-uniform sampling density. Our numerical
experiments show that these improvements in accuracy are realizable in practice
when using $\delta$-nets as spatially uniform inputs to the TMDmap algorithm.