# Questions tagged [optimal-transportation]

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133
questions

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43 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 ...

**1**

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49 views

### Producing a minimiser for the Kantorovich problem from a minimiser of the Beckmann flow problem

Notation: We denote by $\mathcal M$ the set of vector valued measures on $\mathbb R^d$ whose divergence is a scalar measure (in the weak sense).
Definitions: Consider the Beckmann flow minimisation ...

**4**

votes

**1**answer

139 views

### Is the optimal transport of radially symmetric measures also radially symmetric?

Let $\mu$ and $\nu$ be radially symmetric probability measures on $\mathbb R^d$. Consider the Kantorovich optimal transport problem between $\mu$ and $\nu$, with convex, nonnegative cost. Suppose ...

**0**

votes

**1**answer

48 views

### Is integration against an indicator Wasserstein-Continuous

Let $\mathcal{P}_p(X)$ denote the Wasserstein space over a compact metric space $X$, and $1\leq p<\infty$. Fix a non-empty closed subset $C\subseteq X$. Then is the map:
$$
\mathbb{P} \mapsto \...

**3**

votes

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50 views

### Semiconcavity estimate for the squared distance on a compact Riemannian manifold

I am currently reading this paper on the Riemannian structure of the Wasserstein space over a compact Riemannian manifold (my question doesn't concern the Wasserstein metric), specifically Section 4.1,...

**2**

votes

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103 views

### Simplify Kantorovich–Rubinstein duality when distributions share a common marginal

Consider the product of two metrical spaces $X\times Y$ and two probability distributions $\mu$ and $\nu$ on this product space. By the Kantorovich-Rubinstein duality I know that I can write the ...

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96 views

### Wasserstein distance and Monge-Kantorovich-Rubinstein duality

The definition of Wasserstein $p$-distance between two measures $\mu$ and $\nu$ on a Polish space $X$ is given by
$$
W_p^p(\mu, \nu) = \inf_{\gamma \in \Pi(\mu, \nu)} \int_{X \times X} d(x, y)^p\; d\...

**1**

vote

**1**answer

66 views

### Continuity of pushforward operation

Let $X$ and $Y$ be compact metric spaces and let $f,g:X\rightarrow Y$ be $\epsilon$-uniformly close; i.e.:
$$
\sup_{x \in X} d_Y(f(x),g(x))<\epsilon.
$$
Then, are their push-forwards close in ...

**1**

vote

**1**answer

86 views

### Are there any results on concentration bounds of Wasserstein distances between empirical measures?

I know there are concentration bounds on $W(\mu,\hat{\mu})$ where $\mu$ and $\hat{\mu}$ are true and empirical distributions respectively, but is there anything out there on $W(\mu,\nu)$ versus $W(\...

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49 views

### How does one define weak convergence of probability measures in $L^{\infty}(\Omega)$?

I am reading the following article and on page 9/17 (above Eqn (4.9)) the authors state that if $\gamma_{\epsilon_k}|\_G_{\delta}\times \Omega\to \gamma|\_G_{\delta}\times \Omega$ as $\epsilon_k\to 0$ ...

**0**

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**1**answer

86 views

### Parameterization of exponential family

Let $\{\mathbb{P}_{\theta}\}_{\theta}$ be an exponential family of probability measures, all with finite mean. Under what conditions is the parameterization map $\theta\mapsto \mathbb{P}_{\theta}$ ...

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**1**answer

49 views

### Arbitrarily bad rates of convergence in Wasserstein metric

Suppose $W_p(\mu_n,\mu)\to 0$ and $d(E(\mu_n),E(\mu))<r_n$. Here, $W_p$ is the $p$th-order Wasserstein distance (with respect to the metric $d$) and $\mu_n,\mu$ are probability measures on some ...

**4**

votes

**3**answers

182 views

### How to interpret couplings in optimal transport?

Let $\mu$ and $\nu$ be two measures on some (at least measurable) space $X$. In optimal transport theory, Monge's problem to
$$ \text{minimize} \quad \int c(x,T(x))\mu(dx) \quad \text{over measurable ...

**1**

vote

**0**answers

96 views

### Statistical analysis of optimization solution involving Brenier potentials?

I'm reading the paper https://arxiv.org/pdf/1905.10812.pdf where strongly convex approximations to Brenier potentials are approximated.
Let $\mathcal{E}$ be a partition of $\mathbb{R}^{d}$ and $ 0\leq ...

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61 views

### Relationship between Wasserstein projections and metric projections in a linear space

Let $(X,d)$ be a metric space, $x\in X$, and $Y\subset X$ is a closed set. Assume that $X$ is also a real vector space, so that we can form linear combinations over $\mathbb{R}$, but I do not assume ...

**5**

votes

**2**answers

169 views

### Comparison of Information and Wasserstein Topologies

There are many possible metrics one can place on the space of Gaussian probability measures on $\mathbb{R}^n$, with strictly positive definite co-variance matrices. Let's denote this space by $X$.
I'...

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46 views

### Optimal transport mapping between sets with a common subset

What would the optimal transport mapping between two sets with a common subset be? The problem I'm thinking about is the following:
I'm in $\mathbb{C}^n$ and I have two distributions $\mu$ and $\nu$ ...

**0**

votes

**1**answer

49 views

### One question about how to get $\inf_{\nu\in\mathcal{M}(Z)} \sum_{j=0}^N F_j^*(\nu)=-(\sum_{j=0}^N F_j^*)^*(0)$?

In this paper: Matching for teams by G. Carlier and I. Ekeland (2010), https://www.jstor.org/stable/25619994?seq=1#metadata_info_tab_contents: they claim that follows a standard argument in Ekeland ...

**4**

votes

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79 views

### What kinds of gradient-flows on $\mathbb R^d$ preserve the log-concavity of the distribution $\mu_0$ of starting point $x_0$

Let $\mu_0$ be a log-concave distribution on $\mathbb R^d$ and let $f:\mathbb R^d \to \mathbb R$ be $C^2$. Let $x_0$ be sampled uniformly at random from a log-concave distribution $\mu_0$, meaning ...

**2**

votes

**0**answers

56 views

### Error size of universal lower-bound for Wasserstein distance

In the article On a Formula for the L2 Wasserstein Metric between Measures On Euclidean and Hilbert Spaces the authors discuss that for any $\nu,\mu\in\mathcal{P}_2(\mathbb{R}^d)$, the space of Borel ...

**0**

votes

**1**answer

71 views

### The uniqueness of Barycenters in the Wasserstein space

I am reading the paper Barycenters in the Wasserstein space by Martial Agueh and Guillaume Carlier. In Proposition 3.5, they prove the existence and uniqueness of
$$\nu \mapsto \sum_{i=1}^p \frac{\...

**2**

votes

**0**answers

52 views

### $CD(K,N)$ condition for non complete metric measure spaces

That's basically it. I would like to know if it's possible to define the ${\sf CD}(K,N)$ condition for metric measure spaces that are not necessarily complete. The references I have found on this ...

**0**

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**1**answer

229 views

### Integer solution of optimal transport

Let us consider two vectors $\mathbf{a}=(a_1,...,a_n)$ and $\mathbf{b}=(b_1,...,b_m)$ so that each quantity is an integer $a_i,b_j \in \mathbb{N}$. It represents for example supply and demand. Let $\...

**2**

votes

**1**answer

220 views

### Closed-form upper-bounds for Wasserstein distance between finite measures

Let $x_1,\dots,x_n,y_1,\dots,y_n\in \mathbb{R}$ and such that $x_i\neq x_j$ and $y_i\neq y_j$ if $i\neq j$. Let $a,b$ be elements of the probability n-simplex. Define the measures $\mu\triangleq \...

**0**

votes

**1**answer

97 views

### Intersection of projection of sets

Suppose that we have two arbitrary sets $\mathcal{X}$, $\mathcal{Y}$ and are given a function $c : \mathcal{X} \times \mathcal{Y} \rightarrow \mathbb{R}$
Consider the following inequality for ...

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vote

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35 views

### Lax CD(K, $\infty)$ space in the sense of Sturm

In K.T. Sturm's "On the geometry of metric measure spaces. I", Definition 4.5, he introduces a "lax" version of the usual CD(K,$\infty$) lower bound. Namely, one allows for $\...

**3**

votes

**1**answer

159 views

### Invertibility of neural network as operator on Wasserstein space

Question statement: Consider the space of probability measures with finite second moments $P_2(\mathbb{R}^d)$, which is equipped with the Wasserstein-2 distance $W_2$, and the square integrable ...

**1**

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39 views

### Moduli of continuity and Wasserstein differentiability of functions between measures

Let $X=\mathbb{R}^n$; I am also interested in the general case $X$ is a metric space but for simplicity let's focus on Euclidean space. Let $\mathcal{P}(X)$ denote the space of Borel probability ...

**1**

vote

**1**answer

52 views

### Scaling behavior of Wasserstein distances

Let $p>1$ and $\mu\neq \nu$ be two probability measures on $\Omega\subset \mathbb{R}^d$ a bounded set. For $\alpha \geq 0$, we let $$C_\alpha(\mu,\nu) = \inf_\sigma \frac{W_p(\mu+\sigma,\nu+\sigma)}...

**2**

votes

**1**answer

105 views

### Ideas on how to prove Pythagorean identity involving Wasserstein distances?

I conjectured earlier that if $P$ and $Q$ were two probability measures, then we could show
$$W^2(P,Q) = \min_{T} [d^2(P,T_{\#}P) + W^2(T_{\#}P,Q)]$$ where $W^2(P,Q)$ denotes the squared Wasserstein-2 ...

**7**

votes

**1**answer

265 views

### Stability of displacement interpolation in optimal transport

Let $(X,d)$ be a complete separable metric space, and let $(\mathcal{P}_2 (X), W_2)$ be the space of probability measures on $X$ with finite second moments, equipped with the 2-Wasserstein distance. ...

**13**

votes

**2**answers

706 views

### Do distance functionals separate probability measures?

Let $(\Omega,d)$ be a compact metric space and $\mathcal P(\Omega)$ its space of Borel probability measures. Let $D=\{ d_p\mid p\in\Omega\}$ where $d_p(x)=d(p,x)$ be the set of all "distance ...

**1**

vote

**1**answer

85 views

### Wasserstein space with strictly non-positive sectional curvature

Let $(X,d)$ be a (separable, complete) metric space with uniformly strictly non-positive curvature in the sense of Alexandrov, i.e. $(X,d)$ satisfies a $CAT(K)$ inequality for some $K<0$.
Does it ...

**2**

votes

**1**answer

161 views

### About the metrizability of the space of Probability measures $\mathcal{P}(S)$

It is often proved in Books that the space of Probability measures $\mathcal{P}(S)$ on a Polish metric space $(S,\rho)$ endowed with the weak/narrow topology induced by declaring it to be be the ...

**4**

votes

**2**answers

102 views

### Are there alternative regularizations for optimal transport problems besides entropic regularization?

I see that most of the regularization done involves an entropy term.
Has there been any work done on other regularization methods? In particular, I'm wondering if anyone has done a regularization ...

**4**

votes

**0**answers

89 views

### Upper bound $\tau_C := \int_{\|x\| \le 1}(vol(C \cap (x + C))/vol(C))dx$ for a convex body $C \subseteq \mathbb R^n$, by reducing to a ball

Let $C$ be a convex body in $\mathbb R^n$, i.e a bounded convex subset of $\mathbb R^n$ which has nonempty interior, and which is (A) open, or (B) closed (I'm not sure one makes more sense; choose the ...

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63 views

### Metrics on the space of distributions in terms of p.d.fs

If two probability distributions (on the same measure space) are s.t they have p.d.fs and the $L^1$ distance between the p.d.f.s is large, then is there a choice of a ``nice" metric $d_{\rm ...

**4**

votes

**1**answer

159 views

### Optimal transport: find cost function given observed transport

Could you advise me please on what to read on the "inverse" problem: suppose I have a source measure, a target measure and I observe the solution to optimal transport problem -- can I "back out" the ...

**6**

votes

**1**answer

200 views

### Heat flow, decay of the Fisher information, and $\lambda$-displacement convexity

In the whole post I will work in the flat torus $\mathbb T^d=\mathbb R^d/\mathbb Z^d$ and $\rho$ will stand for any probability measure $\mathcal P(\mathbb T^d)$. This question is strongly related to ...

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**0**answers

69 views

### improved regularization for $\lambda$-convex gradient flows

It is well-known that gradient-flows of convex functionals are "parabolic" in some vague sense, and accordingly solutions tend to regularize instataneously. In the abstract context of gradient flows ...

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75 views

### Explicit formula for this distance between positive semi-definite matrices?

Let $A$ and $B$ in $\mathbb{R}^{d\times d}$ be positive semi-definite (psd) matrices and let $d\tau$ be the uniform probability distribution on the unit sphere $\mathbb{S}^{d-1}$ in $\mathbb{R}^d$. I ...

**3**

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141 views

### A new “adversarial” Wasserstein distance?

Let us consider $\mu_1, \mu_2$ and $\mu_3$ three probability measures living on $[0,1]^{k_1}, [0,1]^{k_2}$ and $[0,1]^k$respectively, with $k_1 +k_2=k$. Let us denote by $\Gamma(\mu,\nu)$ the set of ...

**2**

votes

**1**answer

114 views

### Is the Wasserstein kernel positive definite?

Define a point cloud $X=\{x_i\}_{1\leq i\leq n}$, for $x_i\in\mathbb R^d$. Define the Wasserstein kernel as $$W(X,Y)=\max_{T}\frac{1}{n}\sum_{kl}T_{kl}\langle x_k,y_l \rangle$$
where $T$ is any doubly ...

**2**

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78 views

### Prove the equicontinuity of a maximizing sequence

Let $X$ be a compact subset of $\mathbb{R}$ and $c(x_1,x_2,x_3,x_4)$ be a fixed bounded continuous functions on $X^4$. Assume $\mu,\nu$ are probability measures on $X^2$, and $\mu\otimes\nu$ is the ...

**1**

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**1**answer

318 views

### Transport of measure

Let's disintegrate $\mu$ and $\nu$, two probabilities on $\mathbb{R}^{d}$ , according to
$$
\pi_{k} (x_{1},...,x_{d}) = (x_{k},...,x_{d})
$$
We get a family of measures and each measure $\mu_{k,d}^{+...

**0**

votes

**1**answer

173 views

### Computing discrete optimal transport

I am trying to find a combinatorial approach to solve the following optimization problem.
\begin{align}
&\max_{x_{ij}} C_{ij} x_{ij}, \\
&\text{such that},\\
&\sum_{j} x_{ij} \leq r_i~\...

**0**

votes

**1**answer

78 views

### A problem with the dual form of semi-discrete optimal transport

Consider the uniform distribution $\lambda$ on $[0,1]$, and a point measure $\rho$ with density $\frac{1}{2} (\delta_{x_1} + \delta_{x_2})$, where we have $0\le x_1 \le x_2 < 1/2$.
If our cost is ...

**1**

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171 views

### Variational derivative of Wasserstein distance using Benaumou-Brenier formulation

I learned from the gradient flow theory in Wasserstein space that an equation of gradient flow type
$$\partial_t \rho + \nabla \cdot (\rho \nabla \frac{\delta F}{\delta \rho})=0,$$
can be derived as ...

**5**

votes

**1**answer

346 views

### How to control Wasserstein distance in terms of characteristic function

Let $\mathcal P(\Omega)$ be the set of probability measures supported on some compact subset $\Omega\subset\mathbb R^d$. For $\mu\in\mathcal P(\Omega)$, denote by $F_{\mu}$ its characteristic function,...

**3**

votes

**2**answers

412 views

### Maximum cost optimal transport

Kantorovich's optimal transportation problem
\begin{equation}
\tau_c(\mu,\nu)=\min\limits_{\pi\in\Pi(\nu,\mu)} \int_{X\times Y}c(x,y)d\pi(x,y)
\end{equation}
where $\Pi(\mu,\nu) = \{\pi\in P(X\times ...