SIAM/ASA Journal on Uncertainty Quantification, Volume 8, Issue 3, Page 1012-1035, January 2020.
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.

中文翻译：

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不连续密度传输的几何方法

SIAM / ASA不确定性量化期刊，第8卷，第3期，第1012-1035页，2020年1月。
关于输入（“源”）和输出（“目标”）之间关系的不同观察结果经常以直方图（源和目标密度的离散）表示。将这些密度相互传递可以提供有关基本关系的见解。在（前向）不确定性量化中，通常研究系统输入的分布如何影响系统响应的分布。一旦确定了输入和输出的分布，我们将重点放在系统（运输图）本身的识别上，并通过包括所谓的“观察过程”中的数据来建议对当前实践进行修改。我们假设存在关系的平滑流形；然后，源和目标就是该歧管的部分观测结果（可能是投影）。了解这种流形意味着对关系的了解，也就是对源观测值和目标观测值之间的“正确”传输的了解。当源目标观测值不是双射的时（当流形也不是两个观测空间上的函数图时） （因为它们的折叠会引起密度奇异性，或者因为它在几个可观察的事物上边缘化），所以歧管的恢复被遮盖了。使用动力系统中的吸引子重构的思想，我们演示了如何以观察过程的短历史形式显示附加信息帮助我们恢复基础流形。