Interpolating Spline Curves of Measures

Abstract

When dealing with classical, Euclidean data, the statistician's toolkit is enviably deep: from linear and nonlinear regression, to dealing with sparse or structured data, to interpolation techniques, most any problem dealing with vector or matrix data is amenable to several different statistical methods. Yet modern data is often not Euclidean in nature. The semantic content of natural images does not have a vector structure; shifting an image one pixel to the right does not perceptibly change it, but its vector representation is very different. For model cross-validation or bootstrapping, each data point is a dataset in its own right, and one might want to consider an "average dataset''. In an ensemble method, experts may express their beliefs as prior distributions; how would we perform a statistical analysis of these?

Recently much attention has been paid to a framework which subsumes all of these problems: the Wasserstein space of measures with finite second moment. Works on point estimation, generalized means, and linear regression have appeared, as have some on smooth interpolation, greatly expanding the statistical toolkit for modern data. In this vein, the present work is broadly a theoretical and computational exploration of curves of measures which in some sense minimize curvature while interpolating data, as splines do in Euclidean space. We answer several questions about the relationship between the intrinsic Wasserstein-Riemannian curvature of such curves and a particle flow-based, "fluid-dynamical" formulation, and provide fast and accurate smooth interpolation techniques. We also study a related probabilistic interpolation problem unique to the measure setting, which asks for particle trajectories that satisfy certain interpolation constrains and minimize a KL divergence, in analogy with the Schrödinger bridge problem. We conclude with an extension of our methods to the case of unbalanced measures in the Wasserstein-Fisher-Rao space.