TY - JOUR
T1 - A geometric approach to the transport of discontinuous densities
AU - Moosmüller, Caroline
AU - Dietrich, Felix
AU - Kevrekidis, Ioannis G.
N1 - Publisher Copyright:
© 2020 Society for Industrial and Applied Mathematics and American Statistical Association
PY - 2020
Y1 - 2020
N2 - 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 are illustrated and discussed, along with limitations in the recovery of the true underlying relation.
AB - 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 are illustrated and discussed, along with limitations in the recovery of the true underlying relation.
KW - Delay embedding
KW - Discontinuous densities
KW - Manifold reconstruction
KW - Optimal transport
KW - Singularities
UR - http://www.scopus.com/inward/record.url?scp=85093960230&partnerID=8YFLogxK
U2 - 10.1137/19M1275760
DO - 10.1137/19M1275760
M3 - Article
AN - SCOPUS:85093960230
SN - 2166-2525
VL - 8
SP - 1012
EP - 1035
JO - SIAM-ASA Journal on Uncertainty Quantification
JF - SIAM-ASA Journal on Uncertainty Quantification
IS - 3
ER -