Slide 7
Slide 7 text
Optimal transport
What is the optimal way to move or transport the mountain with shape
X, density q(x) to another shape Y with density p(y)? I.e.
DistT
(p, q)2 = inf
T : Ω→Ω Ω
T(x) − x 2q(x)dx: T#
q = p .
The problem was first introduced by Monge in 1781 and relaxed by
Kantorovich in 1940. It introduces a metric function on probability set,
named optimal transport distance, Wasserstein metric or Earth Mover’s
distance (Ambrosio, Gangbo, McCann, Benamou, Breiner, Villani, Otto,
Figali et.al.). Nowadays, optimal transport distances have been shown
useful in inference problems and inverse problems (Poggio, Preye, Yunan,
Engquist, Arjovsky, Osher, et.al.).
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