Is an ambiguity set with Wasserstein distance of order 1 is convex?
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I have a question about the convexity of an Wasserstein ambiguity set.
Let $W_1(mu, nu)$ be Wasserstein distance of order 1 between $mu$ and $nu$ defined as
$$W_1(mu, nu) := minlimits_{gamma in Gamma(mu, nu)} bigg { int_{Xi times Xi} d^p(xi, zeta) gamma(dxi, dzeta) bigg } $$ where $Gamma(mu, nu)$ denote a set of all probability measures on $Xi times Xi$ with marginals $mu$ and $nu$.
Let $nu$ be the empirical distribution. The Wasserstein ambiguity set $mathcal{M}$ is defined by $$mathcal{M} := { mu in mathcal{P}(Xi) : W_1(mu, nu) leq theta }.$$ where $theta$ is given radius.
I am curious about the set $mathcal{M}$ is convex. I notice that Wasserstein distance satisfies the triangle inequality, but I'm not sure that the set $mathcal{M}$ is convex.
Is the Wasserstein ambiguity set of order 1 is convex?
optimization convex-analysis
asked Dec 6 '18 at 0:36
SYLeeSYLee
1
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add a comment |
$begingroup$
I have a question about the convexity of an Wasserstein ambiguity set.
Let $W_1(mu, nu)$ be Wasserstein distance of order 1 between $mu$ and $nu$ defined as
$$W_1(mu, nu) := minlimits_{gamma in Gamma(mu, nu)} bigg { int_{Xi times Xi} d^p(xi, zeta) gamma(dxi, dzeta) bigg } $$ where $Gamma(mu, nu)$ denote a set of all probability measures on $Xi times Xi$ with marginals $mu$ and $nu$.
Let $nu$ be the empirical distribution. The Wasserstein ambiguity set $mathcal{M}$ is defined by $$mathcal{M} := { mu in mathcal{P}(Xi) : W_1(mu, nu) leq theta }.$$ where $theta$ is given radius.
I am curious about the set $mathcal{M}$ is convex. I notice that Wasserstein distance satisfies the triangle inequality, but I'm not sure that the set $mathcal{M}$ is convex.
Is the Wasserstein ambiguity set of order 1 is convex?
optimization convex-analysis
asked Dec 6 '18 at 0:36
SYLeeSYLee
1
$endgroup$
$begingroup$
convexity follows from the triangle inequality and positive homogeneity
$endgroup$
– LinAlg
Dec 6 '18 at 0:52
add a comment |
$begingroup$
I have a question about the convexity of an Wasserstein ambiguity set.
Let $W_1(mu, nu)$ be Wasserstein distance of order 1 between $mu$ and $nu$ defined as
$$W_1(mu, nu) := minlimits_{gamma in Gamma(mu, nu)} bigg { int_{Xi times Xi} d^p(xi, zeta) gamma(dxi, dzeta) bigg } $$ where $Gamma(mu, nu)$ denote a set of all probability measures on $Xi times Xi$ with marginals $mu$ and $nu$.
Let $nu$ be the empirical distribution. The Wasserstein ambiguity set $mathcal{M}$ is defined by $$mathcal{M} := { mu in mathcal{P}(Xi) : W_1(mu, nu) leq theta }.$$ where $theta$ is given radius.
I am curious about the set $mathcal{M}$ is convex. I notice that Wasserstein distance satisfies the triangle inequality, but I'm not sure that the set $mathcal{M}$ is convex.
Is the Wasserstein ambiguity set of order 1 is convex?
optimization convex-analysis
asked Dec 6 '18 at 0:36
SYLeeSYLee
1
$endgroup$
I have a question about the convexity of an Wasserstein ambiguity set.
Let $W_1(mu, nu)$ be Wasserstein distance of order 1 between $mu$ and $nu$ defined as
$$W_1(mu, nu) := minlimits_{gamma in Gamma(mu, nu)} bigg { int_{Xi times Xi} d^p(xi, zeta) gamma(dxi, dzeta) bigg } $$ where $Gamma(mu, nu)$ denote a set of all probability measures on $Xi times Xi$ with marginals $mu$ and $nu$.
Let $nu$ be the empirical distribution. The Wasserstein ambiguity set $mathcal{M}$ is defined by $$mathcal{M} := { mu in mathcal{P}(Xi) : W_1(mu, nu) leq theta }.$$ where $theta$ is given radius.
I am curious about the set $mathcal{M}$ is convex. I notice that Wasserstein distance satisfies the triangle inequality, but I'm not sure that the set $mathcal{M}$ is convex.
Is the Wasserstein ambiguity set of order 1 is convex?
optimization convex-analysis
optimization convex-analysis
asked Dec 6 '18 at 0:36
SYLeeSYLee
1
asked Dec 6 '18 at 0:36
SYLeeSYLee
1
asked Dec 6 '18 at 0:36
SYLeeSYLee
1
asked Dec 6 '18 at 0:36
SYLeeSYLee
1
asked Dec 6 '18 at 0:36
SYLeeSYLee
1
1
$begingroup$
convexity follows from the triangle inequality and positive homogeneity
$endgroup$
– LinAlg
Dec 6 '18 at 0:52
add a comment |
$begingroup$
convexity follows from the triangle inequality and positive homogeneity
$endgroup$
– LinAlg
Dec 6 '18 at 0:52
$begingroup$
convexity follows from the triangle inequality and positive homogeneity
$endgroup$
– LinAlg
Dec 6 '18 at 0:52
$begingroup$
convexity follows from the triangle inequality and positive homogeneity
$endgroup$
– LinAlg
Dec 6 '18 at 0:52
add a comment |
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$begingroup$
convexity follows from the triangle inequality and positive homogeneity
$endgroup$
– LinAlg
Dec 6 '18 at 0:52