Questions tagged [wasserstein]
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65
questions
6
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Can we bound the L1 distance between densities by Wasserstein distance of measures
Let $\mu_1$ and $\mu_2$ be two probability measures over a closed interval $[a, b]$, with respective density functions $\phi_1$ and $\phi_2$. Is there a way to bound the $L^1$ distance of the ...
- 317
5
votes
1
answer
130
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Advection reaction equation is solved by projection of solution of continuity equation
Suppose an absolutely continuous curve $\mu \colon (0, \infty) \to P_2(\Omega)$, where $P_2$ is the Wasserstein-2-space, fulfils the continuity equation
$$ \label{eq:CE} \tag{CE}
\partial_t \mu_t = \...
- 4,878
4
votes
0
answers
52
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Concentration inequalities for random measures
For random variables $X_1,\dots,X_n$ with common mean $\mathbb{E}[X_i]=\mu$ and common bounds $a\leq X_i\leq b$, we have the very useful Hoeffding's inequality:
$$\mathbb{P}\left(\left|\mu -\frac1n\...
- 1,261
4
votes
1
answer
136
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Maximiser of $W_1(\mu, \nu)$ can be changed outside of $\text{conv}(\text{supp}(\mu) \cap \text{supp}(\nu))$ (under additional assumptions)
Let $(X, \| \cdot \|)$ be a reflexive Banach space and $\mathbb{P}_n$, $\mathbb{P}_r$ be measures on $X$.
Let the support of $\mathbb{P}_r$, $M := \text{supp}(\mathbb{P}_r)$ be a weakly compact set ...
- 4,878
3
votes
1
answer
477
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Quantitative bound on Wasserstein distances by $L^p$ distances?
Given two smooth probability densities $f$ and $g$ on $\mathbb{R}$ (or $\mathbb{R}_+$) with finite $p$-th moments. I am wondering if anyone is aware of some explicit upper bound on $W_p(f,g)$ in terms ...
- 2,860
3
votes
1
answer
299
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Boundedness of $\dfrac{W_2(\mu_1+\varepsilon (\mu_2-\mu_1),\mu_1)}{\varepsilon}$ for 2 -Wasserstein metric
Let $\mathcal{P}_2(\mathbb{R}^{n})$ the space of Borel probability measures of finite second moment in $\mathbb{R}^{n}$ equipped with the $2$-Wasserstein metric $W_2$. Let $\mu_1$, $\mu_2 \in \mathcal{...
- 395
3
votes
1
answer
186
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Relations between Kolmogorov-Smirnov distance and Wasserstain distance
Let's have $X,Y \in \mathbb{R}$ with probability measures $\mu, \nu$, then the Kolmogorov-Smirnov distance is defined as follows
$$ d_K(X,Y)=\underset{x \in \mathbb{R}}{sup}\{|F_X(x) - F_Y(x)|\} $$
...
- 155
3
votes
1
answer
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
3
votes
1
answer
148
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Will $L^1\log L^1$ bound gives strong $L^1$ convergence?
I am trying to prove this statement: given a sequence $\{f_n | f_n > 0, c_1 \leq \int_{\Omega} f_n \log{f_n} \leq c_2 \}$, here $\Omega$ is a bounded domain; can we prove the $L^1$ strong ...
- 33
3
votes
1
answer
99
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Covering with sets of negligible boundary
I am studying causality theory in Lorentzian length spaces, and I have a question about geometric measure theory in general (it will help me with a proof I am trying to finish):
Suppose we have a ...
- 441
3
votes
1
answer
123
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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
2
votes
1
answer
810
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Relation between Wasserstein distance and distribution convergence
Let's have a succession $X_n$ of real value random variable and another real value random variable X, then
$$ X_n \overset{d}{\xrightarrow{}} X \iff \lim_{n \to \infty}{}d_K(X_n,X) = 0 $$
where $d_K(X,...
- 155
2
votes
1
answer
58
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Prove $\frac{1}{n}∑_{k=1}^n \Big(x_k^2 + 2\sqrt{3}(1 - \tfrac{2k-1}{n}) x_k + 1 \Big)>0$ given $x_1<…<x_n$
I know for a fact that
$$\frac{1}{n}∑_{k=1}^n \Big(x_k^2 + 2\sqrt{3}(1 - \tfrac{2k-1}{n}) x_k + 1 \Big)>0 \qquad\text{if $x_1<x_2<…<x_n$}$$
should hold because I derived this sum as the ...
- 11.8k
2
votes
1
answer
362
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Wasserstein metric vs Holder continuity
It is well known that if $f$ is a Lipschitz continuous function, i.e.
$$\forall x,y\in \Omega\qquad |f(x)-f(y)|\le L\|x-y\|$$
then, for any two probability distributions $\mu, \nu$
$$\int_\Omega f(x)(...
- 1,199
2
votes
1
answer
449
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Optimal Transport between two Gaussians
Consider the optimal transport map $T$ between $N(\mu_0,\Sigma_0)$ and $N(\mu_1,\Sigma_1)$. I believed that the optimal transport was given by:
$$ T(x) = \mu_1 + \Sigma_1^{1/2} \Sigma_0^{-1/2}(x-\mu_0)...
- 88