Slide 17
Slide 17 text
Explicit formula for discrete ν
Corollary (Point measure target)
Let ν := n
j=1
wj
δxj
. Then Fν flow is given by
[g(t)](s) :=
Qµ0
(s) + 2 (s − Rs,0
) t, t ∈ [ts,0
, ts,1
),
xs,j
+ 2 (s − Rs,j
) (t − ts,j
), t ∈ [ts,j
, ts,j+1
),
Qν
(s), t ≥ ts,|ℓs−ks|
,
where
ts,0
:= 0, ts,1
:=
xs,1
− Qµ0
(s)
2(s − Rs,0)
, ts,j+1
:= ts,j
+
xs,j+1
− xs,j
2(s − Rs,j )
,
Qµ0
(s) ≤ Qν (s) Qµ0
(s) ≥ Qν (s)
ℓs Wℓs−1
< s < Wℓs
Wℓs−1
< s < Wℓs
ks xks
≤ Qµ0
(s) < xks+1
xks−1
< Qµ0
(s) ≤ xks
xs,j xks+j
xks−j
j ≤ |ℓs
− ks
|
Rs,j Wks+j
Wks−j−1
j ≤ |ℓs
− ks
| − 1
Viktor Stein W2 Gradient Flows of MMD functionals with Distance Kernel August 30th, 2024 17 / 22
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µ0