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minh

X

(v;w)=e

d2(hv; hw)

𝑑 ℎ𝑣, ℎ𝑤 𝑑

𝜌𝑣, 𝜌𝑤 ∈ Prob ℝ2

W2(½v; ½w) := inf¼2¦(½v;½w)

µZZ

R2jx¡ yj2 d¼(x; y)

¶1=2

Π(𝜌𝑣, 𝜌𝑤) ℝ2 × ℝ2 𝜌𝑣

𝜌𝑤

𝝆𝒗 𝝆𝒘

𝐺 = (𝑉, 𝐸) 𝑉0 ⊆ 𝑉

𝑣 ∈ 𝑉0

𝑣 ∉ 𝑉0

𝜌: 𝑉 → Prob(𝐷)

ED[½] :=X

(v;w)2E

W22 (½v; ½w)

Prob(ℝ)𝜌𝑣, 𝜌𝑤 ∈ Prob(ℝ)

𝐹𝑣, 𝐹𝑤. 𝑊2 𝜌𝑣, 𝜌𝑤 = 𝐹𝑣−1 − 𝐹𝑤

−1

2

𝑣 ∈ 𝑉0 𝐹𝑣 𝜌𝑣 𝑠 ∈ [0,1]𝑔𝑠: 𝑉 → ℝ

𝛥𝑔𝑠 = 0 ∀𝑣 ∈ 𝑉\𝑉0 𝑔𝑠 𝑣 = 𝐹𝑣−1 𝑠 ∀𝑣 ∈ 𝑉0 𝑣

𝑠 ↦ 𝑔𝑠(𝑣) 𝜌𝑣

𝑣 ↦ 𝜌𝑣

http://realgl.blogspot.com/2013/01/pdf-cdf-inv-cdf.html

[CDF] CDF-1

Linear solve

Prob(ℝ)

•

•

v0 v1v1=2

½0 ½1

v1=4 v3=4

Prob(ℝ)

Fixed

Fixed

Propagated

v0

v1PDF Was. PDF Was.PDF Was.PDF Was.Was.PDF

Prob(𝑆1)

Ground truth (mean)PDF (mean) Wasserstein (mean)

Ground truth (std. dev.)PDF (std. dev.) Wasserstein (std. dev.)

𝑀 𝜙: 𝑉 → 𝑀 𝜌𝜙: 𝑉 →

Prob(𝑀) 𝛿𝜙 𝑣

0

1

2 10 1

𝑆1