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.

翻译：我们获得了目标测度扩散映射（TMDmap）（Banisch 等，2020 年）一致性误差的渐近尖锐误差估计。TMDmap 是扩散映射的一种变体，具有重要性采样功能，因此允许输入数据来自任意密度。推导出的误差估计包括偏差误差和方差误差。所得收敛速度与图拉普拉斯算子的逼近理论一致。我们结果的关键新颖之处在于对所有主导项的前置因子进行了显式量化。我们还证明了使用 TMDmap 获得的 Dirichlet 边值问题解的误差估计，表明解误差受一致性误差控制。我们利用这些结果研究了 TMDmap 在过渡路径理论（TPT）框架下分析过阻尼 Langevin 动力学系统中稀有事件的重要应用。TPT 的核心要素是提交者问题的求解，这是一个针对向后 Kolmogorov 偏微分方程的边值问题。值得注意的是，我们发现由于前置因子公式中若干误差项的抵消，TMDmap 算法特别适合作为提交者问题的无网格求解器。此外，当使用准均匀采样密度时，偏差误差和方差误差会显著改善。我们的数值实验表明，当使用 $\delta$-网作为 TMDmap 算法的空间均匀输入时，这些精度的改善在实际中是可行的。