On a Formula for the L2 Wasserstein Metric between Measures on Euclidean and Hilbert Spaces

By M. Gelbrich. 1990

Subfields: Probabilistic & Bayesian Methods

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In Probabilistic & Bayesian Methods

1. G. Em Karniadakis
2. Liu Yang
3. Nicolas Papernot
4. Soroosh Shafieezadeh-Abadeh
5. Max Sommerfeld
6. Peyman Mohajerin Esfahani
7. Zakhar Shumaylov
8. Viet Anh Nguyen

Citations in the corpus

• The Curse of Recursion: Training on Generated Data Makes Models Forget (2023, background): “A nice property of Wasserstein-2 distance is that Gaussians provide a universal lower bound for the Wasserstein distance [Gelbrich, 1990].”
• Inference for empirical Wasserstein distances on finite spaces (2016, background): “Beyond this case, explicit coupling results are known only for multivariate Gaussian and elliptic distributions (Gelbrich, 1990).”
• An entropic generalization of Caffarelli’s contraction theorem via covariance inequalities (2022, background): “Recall that the Hessian of the Brenier potential between the Gaussian distributions N (0, A) and N (0, B) is the matrix AB [Gel90].”
• Thermodynamic Unification of Optimal Transport: Thermodynamic Uncertainty Relation, Minimum Dissipation, and Thermodynamic Speed Limits (2022, background): “It has been previously proved that W2(p , p) can be bounded from below by the means and covariances of distributions p and p as [170] W2(p , p)(2) ≥ ∥μA − μB∥ 2”
• Statistical Aspects of Wasserstein Distances (2018, background): “For the Gaussian case, see Olkin & Pukelsheim (1982) or Givens & Shortt (1984) in finite dimensions, and Gelbrich (1990) and Cuesta-Albertos et al. (1996) for an infinite-dimensional extension.”
• Wasserstein Distributionally Robust Optimization: Theory and Applications in Machine Learning (2019, background): “To show this, we define the Gelbrich risk as R ( µ, Σ ,(cid:96) ) = sup Q ∈ G ε ( µ, (cid:98) Σ) R ( Q ,(cid:96) ) (18) covariance matrix (cid:98) Σ ∈ S m and if p ≥ 2 , then The representation (17) of the Gelbrich hull as a union of R ε ( (cid:98) µ, (cid:98) Σ ,(cid:96) ) = sup ε (cid:98) (cid:98) ”
• Potential Flow Generator With L2 Optimal Transport Regularity for Generative Models (2019, background): “Suppose μ = N (m1,Σ1), ν = N (m2,Σ2), then [Gelbrich, 1990] the minimum transport cost between μ and ν will be”