Slide 12
Slide 12 text
Gromov Wasserstein between incomparable spaces
Discrete case
cost matrix
cost matrix
μ0
= ∑
i
πi
0
δxi
Cx
μ1
= ∑
j
πj
1
δyj
Cy
GWp
p
(Cx, Cy, μ0
, μ1
) := min
w∈Π(π0
,π1
)
∑
i,j,k,l
Cx
i,k
− Cy
j,l
p
wi,j
wk,l
Cx
i,k
Cy
j,l
i
k
j
l
Non convex quadratic assignment problem, NP-hard
Approximate algos required.
GWp
p
(μ0
, μ1
) := inf
γ∈Π(μ0
,μ1
)
∫
𝒳
×
𝒴
∫
𝒳
×
𝒴
|c
𝒳
(x, x′

) − c
𝒴
(y, y′

)|p dγ(x, y)dγ(x′

, y′

)
[Mémoli 2011, Sturm 2012, Solomon et al. 2016, Vayer et al. 2020 etc…]
If and , , pseudo-metric between metric measure
spaces, i.e. i
ff
isometry s.t.
c
𝒳
= dq
0
c
𝒴
= dq
1
q ≥ 1 GWp
GWp
(μ0
, μ1
) = 0 ∃ψ : (
𝒳
, d0
) → (
𝒴
, d1
) μ1
= ψ#μ0
w?