Questions tagged [optimal-transportation]
The optimal-transportation tag has no usage guidance.
169
questions
1
vote
0
answers
81
views
What is the role of of continuity in this proof of Kantorovich duality?
I'm reading the proof of Kantorovich duality from Villani's book Topics in Optimal Transportation.
Let $X$ and $Y$ be Polish spaces. Let $P(X), P(Y)$ be the spaces of all Borel probability measures ...
0
votes
1
answer
52
views
Why is $\Xi \equiv 0$ if $E=C_{0}(X \times Y)$?
I'm reading the proof of Kantorovich duality from Villani's book Topics in Optimal Transportation. In page 28, the author said that
Exercise 1.11. Let us try to extend this proof to the non-compact ...
1
vote
0
answers
46
views
Sobolev variant of Wasserstein space
Let $\mathcal{P}(\mathbb{R}^n)$ be the set of Borel probability measures on the Euclidean space $\mathbb{R}^n$ and consider thereof consisting of all probability measures $\mathbb{P}$ satisfying $\int\...
1
vote
1
answer
109
views
Fokker-Planck: equivalence between linear spectral gap and nonlinear displacement convexity?
In a smooth, bounded and convex domain $\Omega\subset \mathbb R^d$ consider the usual linear Fokker-Planck equation with Neumann (some would say Robin) boundary conditions
\begin{equation}
\label{FP}
\...
1
vote
0
answers
112
views
Radon-Nikodym derivative of vector-valued measure with respect to another vector-valued measure
Let $(X, | \cdot |)$ be a Banach space.
I am investigating whether one can extend the definition of the Kullback-Leibler divergence
$$
\text{KL}(\mu \ \Vert \ \nu)
:= \int_{\Omega} \ln\left(\frac{\...
1
vote
1
answer
77
views
Given iid samples from the joint distribution $P$ of pair of r.v.'s $(X,Y)$, how to get iid samples from independence coupling $P_X \otimes P_Y$?
Let $(X,Y)$ be a pair of random variables on a measure space $\mathcal T \subseteq \text{"subsets of }\mathbb R^2\text{"}$, with joint probability distribution $P$.
We don't assume $X$ and $Y$ are ...
9
votes
2
answers
202
views
Hyperbolic space embeds into Wasserstein space
Fix a positive integer $n$, let $\mathbb{H}^n$ be the $n$-dimensional hyperbolic space, $r>0$, $x\in \mathbb{H}^n$ and consider the closed (compact) geodesic ball $B_{\mathbb{H}^n}(x,r)$. Are ...
1
vote
0
answers
49
views
Hyperplanes which equalize the Radon transforms of two distributions
Let $p_1$ and $p_2$ be "nice" probability densities on $\mathbb R^m$, for example the densities of a multivariate Gaussians $N(\mu_1,\Sigma)$ and $N(\mu_2,\Sigma)$ with common covariance ...
1
vote
1
answer
47
views
Expectation of a function according to a family of distributions
Consider a family of smooth, atomless CDFs, $F_x(\cdot)$, for each $x \in \mathbb R$. Suppose that $F_x(\cdot)$ are FOSD ranked in $x$. That is, for any $x, x'$ such that $x \ge x'$, $F_x(\cdot) \le ...
0
votes
1
answer
89
views
Measurability of Markov kernel wrt the Borel $\sigma$-algebra generated by the weak topology
Consider two Polish metric probability spaces $(\mathcal{A}, \Sigma_\mathcal{A})$ and $(\mathcal{B}, \Sigma_\mathcal{B})$, endowed with their Borel $\sigma$-algebras. Denote as $\mathcal{P}_\mathcal{B}...
1
vote
0
answers
24
views
Sufficient condition for an $n$-tuple to be a convex conjugate
We say $(f_1,f_2,\dotsc,f_N)$ is a convex conjugate if for any $i=1,2,\dotsc,N$ and any $x_i\in\Bbb R^d$, we have:
$$f_i(x_i)=\sup\left\{\sum_{k=1}^{N}\sum_{j=k+1}^N x_k x_j - \sum_{j=1,j\neq i}^N f_j(...
0
votes
2
answers
134
views
Building the Wasserstein space by pushforwards
Let $\mathbb{R}^d$ denote the $d$-dimensional Euclidean space, $\mathcal{W}_2(\mathbb{R}^d)$ denote the $2$-Wasserstein space with respect to the $d$-dimensional Euclidean space $\mathbb{R}^d$. Let $...
0
votes
0
answers
48
views
Nontrivial lower-bound for $\mathbb P(f(X) \le \alpha \|\nabla f(X)\|)$
Let $f:\mathbb R^d \to \mathbb R$ be a differentiable function whose gradient is $L$-Lipschitz (w.r.t euclidean norm), for some fixed $L \in [0,\infty)$. Let $X=(X_1,\ldots,X_d)$ be a concentrated ...
2
votes
0
answers
65
views
Estimate of Wasserstein distance and flow of vector fields under particular assumptions
Let $\mu$ be a compactly supported absolutely continuous probability measure. Let $v,u$ be Lipschitz vector fields. For a vector field $w$ recall that $\Phi_t^w$ denotes its flow.
A classical estimate ...
1
vote
1
answer
145
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(...
3
votes
1
answer
116
views
Regularity of transport map
Let $\mu$ and $\nu$ be probability measures on $\mathbb{R}^n$ with first moment and suppose that both $\mu$ and $\nu$ have a densities with respect to the $n$-dimensional Lebesgue measure. Fix some ...
1
vote
1
answer
65
views
How to prove the limit of minimizing sequence of measures is again absolutely continuous(w.r.t. Lebesgue) in the minimizing movement scheme?
I am considering the minimizing movement scheme related to the gradient of entropy functional in 2-Wasserstein space. The problem is to minimize the following functional for each fixed $\eta$ which is ...
0
votes
0
answers
69
views
Reference request: optimal transport from Wiener to Poisson
This question arises to me when reading the paper OPTIMAL TRANSPORT FROM LEBESGUE TO POISSON : Let $\mathcal D(\mathbb R_+)$ be the space of right-continuous functions $x:\mathbb R_+\to\mathbb R$ s.t. ...
1
vote
1
answer
144
views
Gradient of Wasserstein distance in the sense of Otto's calculus
I am learning the idea of "gradient" of a functional in Otto's calculus. It is defined as follows.
Suppose the space we are thinking about is $(\mathcal{P}_{2,AC}(\mathbb{R}^d),W_2)$, the ...
0
votes
1
answer
98
views
Uniqueness of maximizer of dual Kantorovich problem with quadratic(or any strictly convex) cost
I am considering the optimal transport problem under the setting $X=\mathbb{R}^n$, $\mu,\nu\in\mathcal{P}(X)$ be two probability measures, and the cost function is $c(x,y)=|x-y|^2$. We know from ...
4
votes
1
answer
126
views
Wasserstein-type concentration inequalities for empirical measures on polish spaces
Let $(\mathcal{X},d)$ be a Polish (metric) space and let $\{X_n\}_{n=1}^{\infty}$ be a sequence of i.i.d. $\mathcal{X}$-valued random elements defined on a common complete (standard) probability space ...
0
votes
2
answers
148
views
Couplings as generalized functions
I've been casually reading about optimal transport, and I was intrigued by the Wasserstein metric, in which we define the distance between two measures $\mu$ and $\nu$ on a metric space $X$ by
$$
W_p(\...
4
votes
2
answers
206
views
Wasserstein convergence of "series expansion'' of probability measure
Let $X$ be a Polish space and let $(\mu_i)_{i=1}^{\infty}$ be a sequence of probability measures in the Wasserstein space $\mathcal{P}(X)$ on $X$. Let $(\beta_i)_{i=1}^{\infty}$ be a summable ...
0
votes
0
answers
37
views
Bounding the Jacobian determinant on the mapping given by Brenier’s Theorem
I have two distributions with small 2-Wasserstein distance between them that satisfy the conditions for Brenier's theorem. I am looking for conditions under which I can prove $p(x)\leq(1+a)\cdot q(x)+...
3
votes
0
answers
46
views
Improving log-Sobolev inequalities via quadratic regularisation
Suppose that $\mu(dx) = \exp(-\psi(x)) \, \mathrm{dx}$ is a probability measure on $\mathbf{R}^d$.
For suitable functions $g \geqslant 0$, define
$$\text{Ent}(g) = \int \mu(dx) g(x) \log \left( \frac{...
3
votes
1
answer
131
views
Open problem 1.28: $W^{1,1}$ regularity for optimal transport map
While I don't work on the regularity theory for the optimal transport map, I was curious about the open problem 1.28 listed in Ambrosio and Gigli's User's guide: the problem to determine whether we ...
1
vote
1
answer
226
views
An approximation problem w.r.t marginal distribution of coordinates of uniform random vector on high-dimensional unit-sphere
Let $X=(X_1,\ldots,X_d)$ be uniformly distributed on the sphere of radius $\sqrt{d}$ in $\mathbb R^d$. Fix a "sufficiently integrable" function $h:\mathbb R \to \mathbb R$, and define ...
0
votes
0
answers
58
views
Empirical estimation of Brenier map from data
Let $f:\mathbb R^d \to \mathbb R$ be a "nice" (say, continuous) function define $A = A_f := \{x \in \mathbb R^d \mid f(x) \ge 0\}$ and $B =B_f:= \{x \in \mathbb R^d \mid f(x) \le 0\}$, and ...
0
votes
2
answers
324
views
Wasserstein distance between $N(0,1/d)$ and the marginal distribution of $x_1$ when $x=(x_1,\ldots,x_d)$ is uniform on the unit-sphere in $R^d$
Let $x=(x_1,\ldots,x_d)$ be uniformly distributed on the unit-sphere in $\mathbb R^d$.
Question.
What is a good upper-bound for Wasserstein distance between $N(0,1/d)$ and the marginal distribution ...
1
vote
1
answer
89
views
$L^p$-barycenters via continuous selectors
Let $\mathbb{P}$ be a Borel probability measure on $\mathbb{R}^n$ with $\int_{x \in \mathbb{R}^n} \|x\|^p\mathbb{P}(dx)<\infty$. For which $p\in [1,\infty)$ (other than $p=2$) is the following ...
5
votes
2
answers
350
views
Textbooks or lecture notes about mean field games
I am looking for a good introductory level textbook (or lecture notes) on mean field games that would be suitable for a graduate course. Ideally, it would include some brief words about optimal ...
6
votes
1
answer
437
views
What is the intuition behind the Kantorovich potential in optimal transport?
From what I currently understand, under certain conditions one may turn the usual Kantorovich problem - a minimisation problem in terms of measures into a maximisation problem in terms of functions. ...
0
votes
0
answers
94
views
Convexity of a set of probability densities
Consider the space of probability densities $(P(\mathbb{R}^d), W_2)$ (probability measures on $\mathbb{R}^d$ with 2-Wasserstein distance).
How can we determine if a subset $Q$ is convex?
I know that a ...
0
votes
1
answer
109
views
Upper bound on the $p$-Wasserstein distance $\mathcal{W}_p(\xi,a\,\xi)$ for some constant $a \neq 0$
Let $\xi$ be a random vector taking values in $\mathbb{R}^d$. Is there an upper bound on the $p$-Wasserstein distance $\mathcal{W}_p(\xi,a\,\xi)$ for some constant $a \neq 0$?
I have seen that if $p=...
1
vote
0
answers
93
views
Orthogonality in Wasserstein tangent space for discrete measures with equal mass
Let say I have $N$ discrete probability measures $(\mu_1,...,\mu_N)$ where each of them has $n$ points in $\mathbb{R}^2$ of equal mass.
Let $P(\mathcal{X})$ be the space of these probability measures ...
2
votes
1
answer
91
views
Reweighting probability measures by convex potentials, and contraction in transport distance
Let $W: \mathbf{R}^d \to \mathbf{R}$ be a convex function such that $\int \exp(-W) = 1$, and define probability measures $\mu_y$ by
$$\mu_y (dx) = \exp( - W (x - y)) \,dx,$$
i.e. each $\mu_y$ is a ...
2
votes
1
answer
112
views
Is there an analogue of transportation-cost inequality under a weighted Log-Sobolev Inequality?
It is known that under the Log-Sobolev Inequality for $\pi$, i.e., if for all $\rho$,
$$H_\pi(\rho):=\int \rho(x)\log\frac{\rho(x)}{\pi(x)}dx \leq \frac{1}{2\beta}\int \rho(x)\left\|\nabla \log\frac{\...
2
votes
0
answers
82
views
Closure of finite support measures in the Wasserstein metric
This is a follow-up question to this question:
Closure Wasserstein for pointmasses
Let $(X,d)$ be a metric space, and let $W_1(X)$ be the space of probability measures $\mu$ on $X$ having finite first ...
1
vote
0
answers
70
views
Producing a minimiser for the Kantorovich problem from a minimiser of the Beckmann flow problem
Notation: We denote by $\mathcal M$ the set of vector valued measures on $\mathbb R^d$ whose divergence is a scalar measure (in the weak sense).
Definitions: Consider the Beckmann flow minimisation ...
4
votes
1
answer
238
views
Is the optimal transport of radially symmetric measures also radially symmetric?
Let $\mu$ and $\nu$ be radially symmetric probability measures on $\mathbb R^d$. Consider the Kantorovich optimal transport problem between $\mu$ and $\nu$, with convex, nonnegative cost. Suppose ...
0
votes
1
answer
60
views
Is integration against an indicator Wasserstein-Continuous
Let $\mathcal{P}_p(X)$ denote the Wasserstein space over a compact metric space $X$, and $1\leq p<\infty$. Fix a non-empty closed subset $C\subseteq X$. Then is the map:
$$
\mathbb{P} \mapsto \...
3
votes
0
answers
58
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
0
answers
212
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-...
4
votes
0
answers
293
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
$$
W_p^p(\mu, \nu) = \inf_{\gamma \in \Pi(\mu, \nu)} \int_{X \times X} d(x, y)^p\; d\...
1
vote
1
answer
116
views
Continuity of pushforward operation
Let $X$ and $Y$ be compact metric spaces and let $f,g:X\rightarrow Y$ be $\epsilon$-uniformly close; i.e.:
$$
\sup_{x \in X} d_Y(f(x),g(x))<\epsilon.
$$
Then, are their push-forwards close in ...
1
vote
1
answer
138
views
Are there any results on concentration bounds of Wasserstein distances between empirical measures?
I know there are concentration bounds on $W(\mu,\hat{\mu})$ where $\mu$ and $\hat{\mu}$ are true and empirical distributions respectively, but is there anything out there on $W(\mu,\nu)$ versus $W(\...
0
votes
0
answers
60
views
How does one define weak convergence of probability measures in $L^{\infty}(\Omega)$?
I am reading the following article and on page 9/17 (above Eqn (4.9)) the authors state that if $\gamma_{\epsilon_k}|\_G_{\delta}\times \Omega\to \gamma|\_G_{\delta}\times \Omega$ as $\epsilon_k\to 0$ ...
0
votes
1
answer
117
views
Parameterization of exponential family
Let $\{\mathbb{P}_{\theta}\}_{\theta}$ be an exponential family of probability measures, all with finite mean. Under what conditions is the parameterization map $\theta\mapsto \mathbb{P}_{\theta}$ ...
0
votes
1
answer
56
views
Arbitrarily bad rates of convergence in Wasserstein metric
Suppose $W_p(\mu_n,\mu)\to 0$ and $d(E(\mu_n),E(\mu))<r_n$. Here, $W_p$ is the $p$th-order Wasserstein distance (with respect to the metric $d$) and $\mu_n,\mu$ are probability measures on some ...
6
votes
3
answers
454
views
How to interpret couplings in optimal transport?
Let $\mu$ and $\nu$ be two measures on some (at least measurable) space $X$. In optimal transport theory, Monge's problem to
$$ \text{minimize} \quad \int c(x,T(x))\mu(dx) \quad \text{over measurable ...