A geometric approach to the transport of discontinuous densities

Different observations of a relation between inputs ("sources") and outputs ("targets") are often reported in terms of histograms (discretizations of the source and the target densities). Transporting these densities to each other provides insight regarding the underlying relation. In (forward) uncertainty quantification, one typically studies how the distribution of inputs to a system affects the distribution of the system responses. Here, we focus on the identification of the system (the transport map) itself, once the input and output distributions are determined, and suggest a modification of current practice by including data from what we call "an observation process". We hypothesize that there exists a smooth manifold underlying the relation; the sources and the targets are then partial observations (possibly projections) of this manifold. Knowledge of such a manifold implies knowledge of the relation, and thus of "the right" transport between source and target observations. When the source-target observations are not bijective (when the manifold is not the graph of a function over both observation spaces, either because folds over them give rise to density singularities, or because it marginalizes over several observables), recovery of the manifold is obscured. Using ideas from attractor reconstruction in dynamical systems, we demonstrate how additional information in the form of short histories of an observation process can help us recover the underlying manifold. The types of additional information employed and the relation to optimal transport based solely on density observations is illustrated and discussed, along with limitations in the recovery of the true underlying relation.