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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
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