All Questions
Tagged with optimal-transportation metric-spaces
13 questions
7
votes
3
answers
340
views
Hyperbolic space embeds into Wasserstein space
Fix a positive integer $n$, let $\mathbb{H}^n$ be the $n$-dimensional hyperbolic space, $r>0$, $x\in \mathbb{H}^n$ and consider the closed (compact) geodesic ball $B_{\mathbb{H}^n}(x,r)$. Are ...
- 195
4
votes
1
answer
2k
views
Earth movers distance (EMD) between two multivariate normals. Is it negative definite distance?
I was looking at the closed form formula for 2-Wassersteins distance for multivariate normal distribution on Wikipedia. https://en.wikipedia.org/wiki/Wasserstein_metric#Normal_distributions
It has a ...
- 141
4
votes
0
answers
197
views
Compactness of the unit ball in the space of Radon measures w.r.t. the Kantorovich-Rubinstein norm
This question was posted previously but has not attracted any responses so I am repharising it in a slightly different language hoping to reach a wider community
Let $(X,d)$ be a pointed metric space ...
- 437
3
votes
1
answer
269
views
Does complete and separable Wasserstein space imply a complete base space?
Also asked on math.SE.
Let $(Z,d)$ be a metric space, and for $p\geq 1$, consider a metric space $(W_p,d_{W_p})$ defined by
The Wasserstein Space $\begin{align}W_p = \{\mu|\mu\textrm{ is a Borel ...
- 305
3
votes
1
answer
151
views
Is this map (from the space of probability densities to the Wasserstein space) Lipschitz?
Let $p \in [1, \infty)$. Let $\mathcal P_p(\mathbb R^d)$ be the space of all Borel probability measures on $\mathbb R^d$ with finite $p$-th moments. Let $D_p$ be the collection of all Borel measurable ...
- 657
2
votes
2
answers
293
views
Optimal transport: the existence of an optimal pair of $c$-conjugate functions
$\newcommand{\diff}{ \, \mathrm d}$
Let
$X,Y$ be Polish spaces,
$\mathcal C_b(X)$ the space of all real-valued bounded continuous functions on $X$,
$\mathcal P(X)$ the space of Borel probability ...
- 825
2
votes
0
answers
95
views
Can we control the Wasserstein metric between $\mu$ and $\nu$ by their moment difference?
Fix $p \in [1, \infty)$. Let $(\mathcal P_p(\mathbb R^d), W_p)$ be the Wasserstein space of all Borel probability measures on $\mathbb R^d$ with finite $p$-th moment. Let $D_p$ be the collection of ...
- 657
2
votes
0
answers
123
views
Closure of finite support measures in the Wasserstein metric
This is a follow-up question to this question:
Closure Wasserstein for pointmasses
Let $(X,d)$ be a metric space, and let $W_1(X)$ be the space of probability measures $\mu$ on $X$ having finite first ...
- 11.1k
2
votes
0
answers
302
views
Simplify Kantorovich–Rubinstein duality when distributions share a common marginal
Consider the product of two metric spaces $X\times Y$, and two probability distributions $\mu$ and $\nu$ on this product space. By the Kantorovich-Rubinstein duality, I can write the Wasserstein-1-...
- 53
1
vote
1
answer
306
views
When are Wasserstein spaces $CAT(\kappa)$?
Let $(X,d)$ be a complete and separable metric space and, for $1\leq p<\infty$, let $(\mathcal{P}_p(X,d),W_p)$ be the $p$-Wasserstein space on $(X,d)$. For which $p$ and $(X,d)$ is $(\mathcal{P}_p(...
- 195
1
vote
1
answer
83
views
Lipschitz approximation of a probability measure with finite $1$-st moment by the ones with finite $p$-th moment
For $p \in [1, \infty)$, let $\mathcal P_p (\mathbb{R^d})$ be the space of Borel probability measures on $\mathbb R^d$ with finite $p$-th moment. We endow $\mathcal P_p (\mathbb{R^d})$ with the ...
- 825
1
vote
1
answer
177
views
Does the topology of Wasserstein space $(\mathcal P_p (E), W_p)$ coincide with the initial topology induced by $\mathcal C_b(E) \cup \{g_p\}$?
Let $(E, d)$ be a Polish space and $\mathcal C_b(E)$ the space of all real-valued bounded continuous functions on $E$. Let $p \in [1, \infty)$ and $\mathcal P_p (E)$ the space of all Borel probability ...
- 825
0
votes
0
answers
55
views
Any useful bases for the topology induced by the $t$-Wasserstein distance?
I am working on $\mathbb R ^d$ equipped with the usual Euclidean metric. I know of one nice base for $\mathcal W _t$, namely: $$\left\{ B_p (r) : r>0, p=\sum_{i=1} ^n \alpha_i \delta_{x_i},\text{ ...
- 291