Questions tagged [optimal-transportation]
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260 questions
3
votes
1
answer
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Reformulation as optimization on probability distributions
This is a "soft" question, in the sense that I'm looking for historical remarks and general commentary rather than a definite answer.
For compact $X \in R^n$ and $f : R^n \to R$ consider the problem
...
- 505
1
vote
1
answer
242
views
Barycenter Map on Wasserstein Space
Let $(X,d)$ be a complete separable metric space, $P_1(X,d)$ be the set of Radon probability measures on $X$ satisfying
$$
P_1(X,d)\triangleq \left\{
\nu:\,(\exists x_0\in X)\, \int_{x\in X} d(x,x_0)d\...
- 1,058
6
votes
1
answer
580
views
Why is it difficult to solve the Monge problem directly?
I'm trying to understand something about the Monge problem. The Monge problem is:
Let $c(x,y): \mathbb{R}^d \times \mathbb{R}^d \rightarrow \mathbb{R}^d$ and $$\mathcal{T}(\mu_1,\mu_2) = \{ T: \...
- 427
8
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1
answer
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How does Otto theory work in this example of Wasserstein a.c. curve of probabilities?
I'm trying to read chapter 8 of the book on gradient flows by Ambrosio-Gigli-Savaré. In this context, I would like to better understand how the theory works for the following specific example. Take ...
- 1,675
2
votes
1
answer
194
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Strong convexity of internal energy with respect to Wasserstein metric
It is well known that the internal energy (see, e.g., Definition 3.32 in and Proposition 3.33 in 1) is geodesically convex with the 2-Wasserstein distance. I was wondering under what condition, the ...
- 5,380
1
vote
0
answers
60
views
Euler-Lagrange Equation for Kantorovich Dual Problem
Given two probability measures $\mu$ and $\nu$, the Kantorovich Dual problem for quadratic cost is to
$$
\text{minimize} \quad \int \phi(x)d\mu + \int \psi(y)d\nu
$$
over pairs $(\phi,\psi)\in L^1(d\...
- 325
1
vote
0
answers
97
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Dependency of the Wasserstein metric on its parameters
Let the population on some region $\Omega\subset\mathbb R^d$ be modeled by a density function $\rho:\Omega\to (0,+\infty)$. Provided $n\ge 1$ food trucks labeled by their capacity $p_1,\ldots, p_n\in (...
CommunityBot
- 1
4
votes
1
answer
596
views
Monge-Kantorovich duality with a $\{0,1\}$ cost function
Consider the usual Monge-Kantorovich transportation problem where $X$ and $Y$ are Polish spaces, $\mu$ and $\nu$ are probability measures on $X$ and $Y$, and $c:X\times Y \to \mathbb{R}^+ \cup \{+\...
- 576
10
votes
1
answer
454
views
Open Questions about Wasserstein Space and PDE
While working on my thesis, I encountered the idea of OMT and started reading some more (like Villani's book). In particular, I came across a PhD thesis by Martial Agueh. I thought it was interesting ...
- 1,277
1
vote
1
answer
163
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Metric 1-current decomposition
I've been reading Paolini-Stepanov arcticle and in section 4, at page 6, they define a metric current from a transport:
$$T_{\eta}(\omega)=\int_{\Theta}[[\theta]](\omega)d\eta(\theta),$$
which ...
- 1,277
1
vote
1
answer
1k
views
Monotonicity, Convexity, and Smoothness of the KL-Divergence between Two Brownian Motions with Different Initializers
We consider the two distributions
$$
p_t = p_0 * N(0, tI),\quad q_t = q_0 * N(0, t I),
$$
where $*$ denotes the convolution between two densities, while $p_0$ and $q_0$ have the same mean and ...
- 6,045
3
votes
2
answers
165
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continuity/ measurablity of optimal transport
given polish space $(X,d)$, consider weak* topology of probability. optimal transport of probability $u,v$ is defined by $\pi(u,v)$ such that $\pi(u,v)$ minimizes:
$\{\int d(x,y) d \pi(dx,dy): \pi \...
- 553
2
votes
1
answer
257
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Reference Request: 2-Wasserstein Metric on Wiener Space
Suppose that X is the subspace of the set of probability measures on the classical Wiener space $C[0,T]$, for some $T>0$, comprised of Gaussian measures.
In the finite-dimensional setting, the ...
- 6,045
4
votes
0
answers
187
views
Transport Distance between Level Sets of a Convex Function
Suppose I have a well-behaved, strictly convex function $f : \mathbf{R}^d \to [0, \infty)$, and assume that $f$ has its unique minimiser at $x = 0$, with $f(0) = 0$.
For $y > 0$, I define the ...
- 801
7
votes
0
answers
433
views
(geodesic) smoothness of f-divergence with respect to the Wasserstein metric
We consider the f-divergence, which takes the form
$$
D_f(P \| Q) = \int_\Omega f\left(\frac{dP}{dQ}\right) dQ.
$$
For example, when $f(t) = t \log t$, we obtain the KL-divergence.
My question is ...
- 1,127
5
votes
0
answers
244
views
Distribution of point knowing target in optimal matching
I am a young PhD student in statistics.
Recently, papers (Ambrosio, Stra and Trevisan; Talagrand; Ledoux to cite but a few) tackled the problem of finding the expected cost in an optimal matching, ...
- 555
2
votes
1
answer
317
views
optimal transport, measurable selection
Let $X = Y = \mathbb{R}^d$ and let $\nu$ be a probability measure on $\mathbb{R}^d$. Consider the collection of probability measure $\pi$ on $X\times Y$ such that $\pi$ has $y$-marginal $\nu$:
$$
\Pi(...
- 42k
4
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0
answers
183
views
Regularity of optimal transport of Gaussian mixtures
In one of the problems that I am working on, I came across the topic of smoothness of optimal transport for Gaussian mixtures. In particular, let $P=P_\theta=\sum_{i=1}^k \frac{1}{k}\mathcal{N}(x| \...
- 161
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0
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103
views
Expectation of maximal Wasserstein distance between empirical distribution and a pdf
Let $P$ be a continuous probability distribution on $R^d$, $X$ the random variable $\sim P$, and $
\hat{X}$ be n i.i.d samples drawn according to $P$. We have another variable $\mu \in S^{d-1}$.
Do ...
- 109
2
votes
0
answers
266
views
Optimal transport between Gaussian mixtures and their centers
I have a question about bounding the Wasserstein loss between a continuous Gaussian mixture and a discrete uniform distribution of its centers. In particular, let $P=\frac 1 k \sum_{i=1}^k \mathcal{N}(...
- 576
3
votes
1
answer
311
views
Kantorovich duality with pseudometrics
The usual framework for the Kantorovich duality in optimal transport theory uses Polish spaces as ground spaces for the distributions that should be transported. Are there results available that ...
5
votes
1
answer
2k
views
Earth mover/Wasserstein distance between a pdf and an empirical distribution
This question is inspired by this much older question:
Convergence of an empirical distribution w.r.t. the Hellinger distance
Let $P$ be a continuous probability distribution on a compact subset of $...
- 3,574
3
votes
1
answer
202
views
Is there a coupling that induces a given coupling via a transition kernel?
Let $X,Y$ be two measurable spaces, $\mu,\nu$ two probability measures on $X$, and $\kappa$ a transition kernel from $X$ to $Y$.
Define $\tilde\mu(dy)=\int_X\kappa(dy|x)\mu(dx)$ and $\tilde\nu(dy)=\...
- 576
10
votes
1
answer
274
views
Cutting a Gaussian in two pieces that are maximally separated in the Wasserstein metric
Denote the standard Gaussian probability measure on $\mathbb R^n$ by $\gamma$. We partition $\mathbb R^n$ into two sets $A$ and $A^c$ such that $\gamma(A) = \gamma(A^c) = 1/2$.
Denote by $\gamma_{A}$...
CommunityBot
- 1
4
votes
2
answers
415
views
Effect of perturbing the atoms of a measure on the Wasserstein distance
Let $(X,d)$ be a metric space, $x_1,\ldots,x_N\in X$ and $x_1',\ldots,x_N'\in X$ be atoms, and $G=\sum_{i=1}^Np_i\delta_{x_i}$, $G'=\sum_{i=1}^Np_i'\delta_{x_i}$, and $G''=\sum_{i=1}^Np_i'\delta_{x_i'}...
- 576
2
votes
0
answers
116
views
A variant of the optimal transport
Let $\mu$, $\nu$ and $\gamma$ be three probability measures on $\mathbb R$. Consider the optimisation problem as follows:
$$\inf_{(X,Y,Z)}~ \mathbb E\big[|Y-Z|^2\big],$$
where the inf is taken ...
- 138
2
votes
1
answer
210
views
Controlling Mean Difference Between Product and Joint Distributions Using Optimal Transportation
Suppose we have nonindependent random variables $X \sim P$ and $Y \sim Q$, where $P$ and $Q$ denote their marginal distributions. We are interested in upper bounding
$$
|\mathbf{E}_{X, Y\sim P \...
- 128k
5
votes
1
answer
396
views
Universal decay rate of the Fisher information along the heat flow
I'm looking for a reference for the following fact: In the torus $\mathbb T^d$ let me denote by $u_t=u(t,x)$ the (unique, distributional) solution of the heat equation
$$
\partial_t u=\Delta u
$$
...
- 5,380
3
votes
1
answer
351
views
Regularity of linearized Monge-Ampere equation
Consider the linearized Monge-Ampere equation $ (p-q)+\nabla\cdot(p\nabla\phi)=0$, where $p$ and $q$ are the density functions of two probability measures, which are supported on a bounded domain $\...
- 325
20
votes
2
answers
2k
views
The geometric median of a triangle
Let $\Omega\subset \mathbb R^n$ be a compact domain of dimension $n$. Define the geometric median on $\Omega$ as the point $m_{\Omega}\in \mathbb R^n$ such that the integral $\int_{\Omega}|x-m_{\Omega}...
CommunityBot
- 1
4
votes
0
answers
757
views
Tangent space and gradient on subspace of Wasserstein space given by finitely supported measures
Let $\mathcal{P}_2(M)$ be the 2-Wasserstein space over some Riemannian manifold $(M,g)$ (connected, complete, and without boundary). Let $\mathcal{FP}_2(M,n)$ be the subspace of probability measures ...
- 1,675
3
votes
0
answers
106
views
Dependency of the Wasserstein distance on the parameter: a differential perspective
Let $\mu(dx)=\sum_{i=1}^np_i\delta_{x_i}(dx)$ and $\nu(dy)=\rho(y)dy$ be two probability measures on $\mathbb R^d$. Consider the $2-$Wasserstein distance below:
$$W_2(\mu,\nu)^2 \quad := \quad \inf_{\...
- 345
3
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0
answers
243
views
Parametric distances on product spaces of measures
Disclaimer: Please excuse my loose language. I'm neither an expert in geometry nor probability. Please ask for clarification if something appears unclear or awkward to you.
Let $X$ be a topological ...
- 6,853
4
votes
1
answer
2k
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Earth movers distance (EMD) between two multivariate normals. Is it negative definite distance?
I was looking at the closed form formula for 2-Wassersteins distance for multivariate normal distribution on Wikipedia. https://en.wikipedia.org/wiki/Wasserstein_metric#Normal_distributions
It has a ...
- 141
8
votes
3
answers
936
views
Question about Wasserstein metric
Let $\mu$ and $\nu$ be two probability measures on $\mathbb R^n$ with finite first moment. Denote by $d:=W_1(\mu,\nu)$, where $W_1(\cdot,\cdot)$ stands for the Wasserstein distance of order $1$.
My ...
- 128k
7
votes
1
answer
438
views
An isoperimetric type of inequality in terms of Wasserstein distance/Optimal transport
Let $A \subset \mathbb{R}^n$ be a region having the same volume as an $n$ dimensional ball $B^n_R$ with radius $R$ centring at the origin.
Isoperimetric inequality says:
$ Vol_{n-1} \partial A \geq ...
- 36
4
votes
1
answer
351
views
Inf of Jensen's inequality
I'm reading a monograph that considers the following problem:
$$\inf_{z(t) \in C^1} \int_0^1 c\bigg(\frac{dz(t)}{dt}\bigg) dt\\ z(0) = x, z(1) = y$$
Here $c$ is a convex function, $z(t)$ are paths ...
- 29.8k
7
votes
2
answers
3k
views
The largest Wasserstein distance to uniform distribution among all probability distributions with uniform marginals
I am looking for the largest Wasserstein distance to the uniform distribution among all probability distributions with uniform marginals.
More specifically, let $\Xi=\{1,2,\ldots,N\}^2$, and let $\nu$...
- 1
2
votes
1
answer
309
views
Density in Wasserstein space
I am wondering whether the following result is true:
Let $\mathcal W_p(\mathbb R^d)$ be the Wasserstein space of order $p$ and let $\eta$ and $\gamma$ be two probability measures in $\mathcal W_p(\...
- 121
4
votes
0
answers
589
views
Optimal transport between two distributions in a Markov chain
In a previous question, given an ergodic Markov chain, I'm interesting in sampling as short a path as possible with prescribed distributions for its endpoints. In a comment, I propose that the ...
- 6,853
3
votes
1
answer
752
views
Wasserstein convergence of conditional measures
Suppose $W_r(\mu_n,\mu)\to0$, where $\mu_n$ and $\mu$ are discrete probability measures on some metric space $\Omega$, and that all measures have the same number of atoms $d$ (but not the same atoms):
...
- 710
3
votes
2
answers
758
views
Multi-marginal optimal transport
The notion of Wasserstein distance between two probability measures is well-studied and well-motivated in many different branches of math and stat.
Let $\mu$ and $\nu$ be any two probability measures ...
- 128k
7
votes
1
answer
865
views
Fenchel-Rockafellar Duality in Villani's Book
Villiani writes (some notation changed) in Topics in Optimal Mass Transportation:
Theorem 1.9. Let $E$ be a normed VS, $E^*$ it topological dual. $\Theta$ and $\Psi$ are two convex functions on $E$ ...
- 1,741
2
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0
answers
75
views
existence of minimizer to the dual problem of a martingale optimal transport type problem
Let $\nu$ be a given probability measure on $\mathbb R^2$ and consider function of the following form:
$$L(f)(x_1,x_2)=\sup_{y\,=\,(y_1,y_2)\,\in\, \operatorname{Graph} (f)} \{ x_1 y_2 + x_2 y_1 - ...
- 325
1
vote
1
answer
218
views
How to sample a path between 2 states in a Markov chain
Given an ergodic Markov chain and 2 states $x$ and $y$, how may one algorithmically sample a path (of finite length) $x =: s_0 \rightarrow s_1 \rightarrow \ldots \rightarrow s_T = y$ between $x$ and $...
- 6,853
3
votes
0
answers
362
views
Second-Order Taylor Expansion of Wasserstein Metric and Related Metrics
Suppose that we have a parametric distribution $P_{\theta}$, which is indexed by the parameter $\theta \in \mathbb{R}^d$. Let $W\{\cdot,\cdot\}$ be the Wasserstein Metric between two distributions.
...
- 1,127
2
votes
1
answer
148
views
Fast algorithm for large-scale, asymmetric transportation linear program
I have a large-ish instance of a transportation problem that is very asymmetric, say of dimensions $100\times10000$. I am currently solving it with a stock LP solver, but obviously something like the ...
CommunityBot
- 1
1
vote
1
answer
129
views
Optimization problem restricted to a smaller field?
Let $c:\mathbb R^2\to\mathbb R$ be a Lipschitz and bounded function (which can be supposed as "nice" as possible). Let $\mu$ and $\nu$ be two probability measures on $\mathbb R$ with finite first ...
CommunityBot
- 1
1
vote
0
answers
48
views
Negative Definiteness of Hopf-Lax-Oleinik Semigroup
Denote by $H_{t}$ the Hopf-Lax semigroup, i.e.\begin{equation}
H_{t}f(x)=\inf_{y\in\mathbb{R}}\left\lbrace f(y)+\frac{(x-y)^{2}}{2t}\right\rbrace.\end{equation} Is $H_{t}$ negative definite on bounded,...
- 289
1
vote
1
answer
685
views
Existence of optimal coupling in optimal transport
Let $P,Q$ be any two distributions over a space $\mathcal{X}$ and let $\mathcal{M}(P,Q)$ be the set of all couplings of $P$ and $Q.$ For a given metric $d$ over $\mathcal{X},$ the optimal transport ...
