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

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.

Emergence of the SVD as an interpretable factorization in deep learning for inverse problems

We demonstrate the emergence of weight matrix singular value decomposition (SVD) in interpreting neural networks (NNs) for parameter estimation from noisy signals. The SVD appears naturally as a consequence of initial application of a descrambling transform - a recently-developed technique for addressing interpretability in NNs \cite{amey2021neural}. We find that within the class of noisy parameter estimation problems, the SVD may be the means by which networks memorize the signal model. We substantiate our theoretical findings with empirical evidence from both linear and non-linear settings. Our results also illuminate the connections between a mathematical theory of semantic development \cite{saxe2019mathematical} and neural network interpretability.