All Questions
Tagged with wasserstein functional-analysis
5
questions
1
vote
2
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156
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Wasserstein Metric Inequality
This is the exercise:
This exercise shows that “spreading out” probability measures makes them closer together. Define the convolution of a measure by: for any probability density function $\phi$, let ...
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3
votes
1
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Scaling property of the Wasserstein metric
I would need help with this example.
Let $(S, ||\cdot||)$ denote a normed vector space over $K =\mathbb R$ or $K =\mathbb C$. Let $X$ and $Y$ be $S$-valued random vectors with $E~[~||X||~] < \infty$...
- 61
3
votes
1
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222
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Absolutely continuous curves in Wasserstein distance and measurability.
Let $(X, d, \mu)$ be a metric measure space. Let $P^1(X)$ denote the space of probability measures on $(X,d)$, which have finite first moments, that is:
\begin{equation}
\nu \in P^1(X) \implies \int d(...
- 313
1
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0
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54
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Derivative of Kantorovic Potential wrt to Measure
The Kantorovich Dual of the 1-Wasserstein distance $W_1(p,q)$ between two densities $p(x), q(x)$ is given by
$$W_1(p,q) = \sup_{|f|_L\leq 1} \int f(x)(p(x)-q(x))dx$$
with $|f|_L$ denoting the ...
1
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1
answer
100
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Extension of Kantorovich-Rubinstein inequality.
Let $(\mathcal{X}, \Sigma)$ be a Polish metric space, endowed with the Borel $\sigma$-algebra. Let $\mathscr P$ be the space of probability measures on $\mathcal X$ and $\mathscr P^1$ be defined as
$$\...
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