The Wasserstein Distances Between Pushed-Forward Measures with\n Applications to Uncertainty Quantification

2019/02/14 by Amir Sagiv, Sagiv, Amir · 1 citation
Decision Sciences · #28A10 #60A10 #65D99 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Numerical Analysis (math.NA) #Probabilistic and Robust Engineering Design #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.1902.05451

openalex publication_date 2019/02/14 · openalex created_date 2022/07/29 · openalex updated_date 2026/07/28

Abstract

In the study of dynamical and physical systems, the input parameters are\noften uncertain or randomly distributed according to a measure varrho. The\nsystem's response f pushes forward varrho to a new measure f\∘\n varrho which we would like to study. However, we might not have access to f\nbut only to its approximation g. We thus arrive at a fundamental question --\nif f and g are close in L^q, does g\∘ varrho approximate f\∘\n varrho well, and in what sense? Previously, we demonstrated that the answer\nto this question might be negative in terms of the L^p distance between\nprobability density functions (PDF). Here we show that the Wasserstein metric\nis the proper framework for this question. For any p\≥ 1, we bound the\nWasserstein distance W_p (f\∘ varrho , g\∘ varrho) from above by\n\‖f-g\‖_q. Furthermore, we provide lower bounds for the cases of p=1,2.\nFinally, we apply our theory to the analysis of common numerical methods in the\nfield of computational uncertainty quantification.\n

Cited by

• The Evolution of Pointwise Statistics in Hyperbolic Equations with Random Data

Related

• Efficient Uncertainty Propagation with Guarantees in Wasserstein Distance
• Wasserstein Distance, Fourier Series and Applications
• Bounding Kolmogorov distances through Wasserstein and related integral probability metrics
• Transport and Interface: an Uncertainty Principle for the Wasserstein distance
• Inference for Projection-Based Wasserstein Distances on Finite Spaces