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We obtain asymptotically sharp error estimates for the consistency error ofthe Target Measure Diffusion map (TMDmap) (Banisch et al. 2020), a variant ofdiffusion maps featuring importance sampling and hence allowing input datadrawn from an arbitrary density. The derived error estimates include the biaserror and the variance error. The resulting convergence rates are consistentwith the approximation theory of graph Laplacians. The key novelty of ourresults lies in the explicit quantification of all the prefactors onleading-order terms. We also prove an error estimate for solutions of DirichletBVPs obtained using TMDmap, showing that the solution error is controlled byconsistency error. We use these results to study an important application ofTMDmap in the analysis of rare events in systems governed by overdampedLangevin dynamics using the framework of transition path theory (TPT). Thecornerstone ingredient of TPT is the solution of the committor problem, aboundary value problem for the backward Kolmogorov PDE. Remarkably, we findthat the TMDmap algorithm is particularly suited as a meshless solver to thecommittor problem due to the cancellation of several error terms in theprefactor formula. Furthermore, significant improvements in bias and varianceerrors occur when using a quasi-uniform sampling density. Our numericalexperiments show that these improvements in accuracy are realizable in practicewhen using $\delta$-nets as spatially uniform inputs to the TMDmap algorithm.

Sharp error estimates for target measure diffusion maps with applications to the committor problem