Wasserstein 1-distance of push-forward measures The 2019 Stack Overflow Developer Survey Results Are InWasserstein metric: conditions for the existence of minimizer and dualityDefining a push-forward/ image measure operatorPush-forward measure's Radon-Nikodim DerivativeThe Wasserstein distance on $mathbbR$Wasserstein distances metrize weak convergenceIs the expectation value Lipschitz for the Wasserstein metric?Proof that Lukaszyk-Karmowski metric upper bound Wasserstein metricEstimate w.r.t the Wasserstein DistanceCharacterization of Wasserstein convergenceRelative Entropy and the Wasserstein distance
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Wasserstein 1-distance of push-forward measures
The 2019 Stack Overflow Developer Survey Results Are InWasserstein metric: conditions for the existence of minimizer and dualityDefining a push-forward/ image measure operatorPush-forward measure's Radon-Nikodim DerivativeThe Wasserstein distance on $mathbbR$Wasserstein distances metrize weak convergenceIs the expectation value Lipschitz for the Wasserstein metric?Proof that Lukaszyk-Karmowski metric upper bound Wasserstein metricEstimate w.r.t the Wasserstein DistanceCharacterization of Wasserstein convergenceRelative Entropy and the Wasserstein distance
$begingroup$
Suppose you are given two metric spaces $(X,d_X)$ and $(Y,d_Y)$ and a map $f:X to Y$.
Furthermore take two measures $mu , nu$ in $P_1(X)$ the Wasserstein 1-space over X. Let $gamma in Lambda(mu , nu)$ be a coupling.
I have seen that $(ftimes f)_*gamma$ is a coupling of $f_*mu$ and $f_*nu$, but I am confused about how exactly to write down a proof of this...
I thougth about the following. If A is in the borel $sigma$-algebra of $X$ we have:
$$(ftimes f)_*gamma(Atimes X) = (ftimes f)_*mu(A)=f_*mu (A)$$
and then doing the same with $nu$, but I am suppose thats not correct since $(ftimes f)_*mu$ may not be defined at all...
A second question is about the Wasserstein 1-distance of this push-forward measures.
We have seen in class that$$int_Ytimes Y d_Y(y_1,y_2)(d(ftimes f)_*gamma)(y_1,y_2) = int_Xtimes X d_Y(f(x_1),f(x_2))dgamma(x_1,x_2).$$
I would like to proof this equation but I have no idea which argument I should use...
Thanks for your help!
measure-theory geometric-measure-theory
asked Mar 30 at 10:37
GMTGMT
12
$endgroup$
add a comment |
$begingroup$
Suppose you are given two metric spaces $(X,d_X)$ and $(Y,d_Y)$ and a map $f:X to Y$.
Furthermore take two measures $mu , nu$ in $P_1(X)$ the Wasserstein 1-space over X. Let $gamma in Lambda(mu , nu)$ be a coupling.
I have seen that $(ftimes f)_*gamma$ is a coupling of $f_*mu$ and $f_*nu$, but I am confused about how exactly to write down a proof of this...
I thougth about the following. If A is in the borel $sigma$-algebra of $X$ we have:
$$(ftimes f)_*gamma(Atimes X) = (ftimes f)_*mu(A)=f_*mu (A)$$
and then doing the same with $nu$, but I am suppose thats not correct since $(ftimes f)_*mu$ may not be defined at all...
A second question is about the Wasserstein 1-distance of this push-forward measures.
We have seen in class that$$int_Ytimes Y d_Y(y_1,y_2)(d(ftimes f)_*gamma)(y_1,y_2) = int_Xtimes X d_Y(f(x_1),f(x_2))dgamma(x_1,x_2).$$
I would like to proof this equation but I have no idea which argument I should use...
Thanks for your help!
measure-theory geometric-measure-theory
asked Mar 30 at 10:37
GMTGMT
12
$endgroup$
add a comment |
$begingroup$
Suppose you are given two metric spaces $(X,d_X)$ and $(Y,d_Y)$ and a map $f:X to Y$.
Furthermore take two measures $mu , nu$ in $P_1(X)$ the Wasserstein 1-space over X. Let $gamma in Lambda(mu , nu)$ be a coupling.
I have seen that $(ftimes f)_*gamma$ is a coupling of $f_*mu$ and $f_*nu$, but I am confused about how exactly to write down a proof of this...
I thougth about the following. If A is in the borel $sigma$-algebra of $X$ we have:
$$(ftimes f)_*gamma(Atimes X) = (ftimes f)_*mu(A)=f_*mu (A)$$
and then doing the same with $nu$, but I am suppose thats not correct since $(ftimes f)_*mu$ may not be defined at all...
A second question is about the Wasserstein 1-distance of this push-forward measures.
We have seen in class that$$int_Ytimes Y d_Y(y_1,y_2)(d(ftimes f)_*gamma)(y_1,y_2) = int_Xtimes X d_Y(f(x_1),f(x_2))dgamma(x_1,x_2).$$
I would like to proof this equation but I have no idea which argument I should use...
Thanks for your help!
measure-theory geometric-measure-theory
asked Mar 30 at 10:37
GMTGMT
12
$endgroup$
Suppose you are given two metric spaces $(X,d_X)$ and $(Y,d_Y)$ and a map $f:X to Y$.
Furthermore take two measures $mu , nu$ in $P_1(X)$ the Wasserstein 1-space over X. Let $gamma in Lambda(mu , nu)$ be a coupling.
I have seen that $(ftimes f)_*gamma$ is a coupling of $f_*mu$ and $f_*nu$, but I am confused about how exactly to write down a proof of this...
I thougth about the following. If A is in the borel $sigma$-algebra of $X$ we have:
$$(ftimes f)_*gamma(Atimes X) = (ftimes f)_*mu(A)=f_*mu (A)$$
and then doing the same with $nu$, but I am suppose thats not correct since $(ftimes f)_*mu$ may not be defined at all...
A second question is about the Wasserstein 1-distance of this push-forward measures.
We have seen in class that$$int_Ytimes Y d_Y(y_1,y_2)(d(ftimes f)_*gamma)(y_1,y_2) = int_Xtimes X d_Y(f(x_1),f(x_2))dgamma(x_1,x_2).$$
I would like to proof this equation but I have no idea which argument I should use...
Thanks for your help!
measure-theory geometric-measure-theory
measure-theory geometric-measure-theory
asked Mar 30 at 10:37
GMTGMT
12
asked Mar 30 at 10:37
GMTGMT
12
asked Mar 30 at 10:37
GMTGMT
12
asked Mar 30 at 10:37
GMTGMT
12
asked Mar 30 at 10:37
GMTGMT
12
12
add a comment |
add a comment |
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