Calculation of the Wasserstein Distance Between Probability Distributions on the Line

By S. S. Vallender. 1974

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

Citations in the corpus

On the empirical estimation of integral probability metrics (2012, neutral): “2) is known for (S, ρ(x, y)) = (R, |x− y|) [47, 48], and given as W1(P,Q) = ∫”
Hilbert Space Embeddings and Metrics on Probability Measures (2009, background): “β is primarily of theoretical interest for which we do not know a closed form, whereas a closed form expression to compute W is known only for R (Vallander, 1973).”
Neural Temporality Adaptation for Document Classification: Diachronic Word Embeddings and Domain Adaptation Models (2019, background): “Wasserstein distance or Earth Mover’s distance (Vallender, 1974) measures the distribution differences between source and target domains (Shen et al.”
Measuring Sample Quality with Kernels (2017, methodology): “…properties of the IMQ KSD and to compare its behavior with that of two preexisting discrepancy measures, the Wasserstein distance d W (cid:107)·(cid:107) 2 , which can be computed for simple univariate targets (Vallender, 1974), and the spanner graph Stein discrepancy of Gorham & Mackey (2015).”
Distributionally Robust Stochastic Optimization with Wasserstein Distance (2016, unknown): “(44) By the equivalent definition of one-dimensional Wasserstein distance [43], for η = ∑Mi i=1 δξi m , we have that W1(η , η̂) = minσ ∑Mi m=1 |ξ m − ξ̂ σ(m)|, where the minimum is taken over all Mi-permutations.”
Spatially coupled ensembles universally achieve capacity under belief propagation (2012, background): “The equivalence of the ﬁrst and second alternative descriptions is shown in [111] · of generality assume that ˜ f (0) = 0 .”
Cross-modal Ambiguity Learning for Multimodal Fake News Detection (2022, unknown): “The cross-modal ambiguity can be measured by the similarity, or the distance, between unimodal distributions using methods such as KL divergence and Wasserstein distance [29].”
Measuring Sample Quality with Diffusions (2016, methodology): “) kernel herding [51]. The quality judgments of the Langevin spanner Stein discrepancy(D1)closely mimic those of the L1 Wasserstein distance d W kk , which is computable for simple univariate targets [87]. Each Stein discrepancy was computed in under 0:03s. Under both diagnostics and as previously o”
McKean–Vlasov SDEs under measure dependent Lyapunov conditions (2018, unknown): “Indeed fix μ, ν ∈ Pp(R) and observe that |ESμ(α)− ESν(α)| ≤ 1 α ∫ α 0 |F μ (s)− F ν (s)| ds ≤ 1 α ∫ 1 0 |F μ (s)− F ν (s)| ds = 1 α W1(μ, ν) ≤ 1 α Wp(μ, ν) , where the equality above follows from [35] and an obvious geometric consideration.”
Approximations for Probability Distributions and Stochastic Optimization Problems (2011, unknown): “dKA(P, P̃) = ∫ |GP (u)− GP̃ (u)| du = ∫ |G−1 P (u)− G−1 P̃ (u)| du, where G−1 P (u) = sup{v : GP (v) ≤ u} (see Vallander (1973)).”
Deep Learning on Graphs: A Survey (2018, unknown): “Unlike the existing works that had adopted KL-divergence as the measurement, DVNE used the Wasserstein distance [114] to preserve the transitivity of the nodes similarities.”
Validation of imprecise probability models (2009, background): “…2 with the realizationbased comparison described in section 3 is to recognize that both are essentially special cases of the Wasserstein distance (Vallender 1973; Dobrushin 1970) YX SnY FX −Einf ~ ~ , where the E denotes the expectation operator, and the infimum is taken over all possible random…”