A variational principle for model-based interpolation

Given a multidimensional data set and a model of its density, we consider how to deene the optimal interpolation between two points. This is done by assigning a cost to each path through space, based on two competing goals|one to interpolate through regions of high density, the other to minimize arc length. From this path functional, we derive the Euler-Lagrange equations for extremal motion; given two points, the desired interpolation is found by solving a boundary value problem. We show that this interpolation can be done eeciently, in high dimensions, for Gaussian, Dirichlet, and mixture models.