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Questions tagged [optimal-transport]

For questions related to the theory, applications, and computational aspects of optimal transport and related topics such as the Wasserstein (and other transportation cost) distances, the Monge-Ampere equation, metric gradient flows, martingale optimal transport, and optimal matching.

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Application of Gronwall's lemma to the expectation

Let $X_t$, $Y_t$ be real-valued continuous stochastic processes with finite second moments such that $$ E |X_t-Y_t|^2 \leq \int_0^t K(s) \left( E |X_s - Y_s |^2 + W_2^2( \mu_s, \nu_s) \right) ds, \...
Holden's user avatar
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Bound on the Wasserstein distance

Let $\mu_1$ and $\mu_2$ be probability distributions on $( \mathbb{R}, \mathcal{B}(\mathbb{R}))$ with finite second moments. Let the symbol $C( \cdot, \cdot )$, with placeholders for two distributions,...
Harry's user avatar
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There is a unique coupling between a probability distribution $\mu$ and a degenerate distribution $\mu_0$. [duplicate]

By degenerate distribution I intend https://en.wikipedia.org/wiki/Degenerate_distribution. I cannot see why the set of couplings between some probability measure and a constant would be a singleton. ...
Eloy Mósig's user avatar
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Finding an explicit optimal map using Brenier's Theorem

I have the following set up: $$ X, Y = [-1,1] \\ \text{Probability Measures } \mu, \nu \text{ with densities:} \\ f(x)= \frac{15}{4}x^2(1-x^2) \\ g(y)=\frac{3}{4}(1-y^2) \\ \text{Cost function: } c(x,...
rainingbricks2000's user avatar
2 votes
1 answer
93 views

How to deal with ill conditioned optimization problem (Sinkhorn barycenter)

Problematic (Debiased Sinkhorn barycenter, proposed by H.Janti et al.): Let $\alpha_1, \ldots, \alpha_K \in \Delta_n$ and $\mathbf{K}=e^{-\frac{\mathrm{C}}{\varepsilon}}$. Let $\pi$ denote a sequence ...
Tung Nguyen's user avatar
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1D-Wasserstein-Barycenter closed form

I just want to verify that my reasoning here is correct. It feels very basic but I can't seem to find this result in any textbook. It is well-known that the 1-Wasserstein distance in 1D can be ...
KP4's user avatar
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5 votes
1 answer
120 views

On the Optimal Transport Distance in Equivalent Martingale Measures

I've been working on this idea for a while and could use some guidance. Right now I'm wondering if the formulas I am writing out make sense. Any help is very much appreciated. Problem Statement Idea ...
jeffery_the_wind's user avatar
1 vote
0 answers
22 views

Optimal transport - Monge plan - measure theoretic calculation

In my Optimal transport course we dealt with the Monge optimal transportation problem. We have: some open sets $X, Y \subset \mathbb{R}^d$, a cost function $c : X \times Y \rightarrow [0, +\infty ]$....
Len's user avatar
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1 answer
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Bound for expected value under Wasserstein metric

I'm reading a paper and the following result is presented: $$ (\mathbb{E}_{F}[\|\mathbf{X}\|^k])^{1/k} \leq (\mathbb{E}_{F_{0}}[\|\mathbf{X}\|^k])^{1/k} + \epsilon, \ \forall F\in\mathcal{B}_{p}(F_{0},...
rcescon's user avatar
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Regularity of Kantorovich potentials for general cost function

I know De Philippis and Figalli have a paper studying the regularity of the Kantorovich potential. In Theorem 3.3, the authors show that the potential is $C^{k+2, \beta}$ if the density functions of ...
tianer555's user avatar
1 vote
1 answer
54 views

Most optimal way to transport objects from A to B with multiple workers

Disclaimer: I don't have any formal education in STEM by any mean and just want to clear up an argument I had with some of my coworkers To provide some context, me and some of my coworkers had to move ...
MrSteak's user avatar
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1 vote
0 answers
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Random walk metropolis within Gibbs

I tried to implement the Random Walk Metropolis within Gibbs algorithm from Marie Therese Wolfram's article on Inverse Optimal Transport, but I feel like there's an error. In the algorithm, they ...
user600785's user avatar
2 votes
1 answer
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Prove that if $p \le q$ then $W_p(\mu,\nu) \le W_q(\mu,\nu)$, where $W_p$ denote the Wasserstein distance

Given two numbers $p,q$ such that $1 \le p \le q$. Prove that \begin{align*} W_p(\mu,\nu) \le W_q(\mu,\nu), \end{align*} where $W_p$ denotes the Wasserstein distance of order $p$. My solution: Given ...
Tung Nguyen's user avatar
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2 votes
1 answer
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Extracting a subsequence which converges in the $t$-Wasserstein distance?

Posted this to MathOverflow as well. Assume that $\mu_n$ are probability measures on $\mathbb R ^d$ with finite moments of order $t$, and $\mu_n\to\mu$ weakly. Clearly, $\int |x|^t d\mu_n(x)$ is a ...
J.R.'s user avatar
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Does complete and separable Wasserstein space imply the completeness of the base space?

Also asked on MathOverflow. 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 ...
Kaira's user avatar
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2 votes
1 answer
25 views

$(n + 1)$-cyclical monotonicity implies cyclical monotonicity in $\mathbb{R}^n$?

Say a function $f : \mathbb{R}^n \to \mathbb{R}^n$ is $N$-cyclically monotone if for any $x_1, \dots, x_{N + 1} \in \mathbb{R}^n$ with $x_1 = x_{N + 1},$ it holds that \begin{align} \sum_{i = 1}^{...
Paruru's user avatar
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Showing that a minimzing sequence for the Wasserstein Variance functional must be tight

I am trying to understand this 2011 paper by Agueh & Carlier [https://www.ceremade.dauphine.fr/~carlier/AC_bary_Aug11_10.pdf] where they introduce the notion of barycenter in the 2-Wasserstein ...
dcgentile's user avatar
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27 views

"Exact" definition of $\lambda$-displacement convexity

I am trying to understand the concept of $\lambda$-displacement convexity in optimal transport. However, I met different definitions of it and felt a bit confused. Generally, geodesic is not ...
Onepunch Man's user avatar
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0 answers
35 views

Application of Brenier Theorem

Brenier Theorem says that, if the cost function is quadratic and the probability measure $\mu$ is absolutely continuous with respect to the Lebesgue measure $L$, then there exists a unique gradient of ...
nimaba99's user avatar
1 vote
0 answers
44 views

Similarity between two optimal transport plans [closed]

I have recently been studying the theory of computational optimal transport and am very interested in how to describe the similarity between two optimal transport plans. Specifically, suppose $P_1,Q_1,...
Mzxr's user avatar
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0 answers
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Prove that the primal DRO problem can be rewritten without the optimality on the transport distance

I'm trying to understand a result in optimal transport. Specifically, consider the Distributionally Robust Optimization problem given by $$\sup \left\{\int f d\mu : d_{c}(\mu, \nu) \leq \delta\right\}$...
rcescon's user avatar
  • 286
2 votes
1 answer
87 views

Geodesic Wasserstein space => the base space is also geodesic?

Let $(Z,d)$ be a Polish 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 probability measure on Z ...
Kaira's user avatar
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0 answers
22 views

Definition of marginal probability measures using test function spaces

This is a question from Villani's Topics in Optimal Transportation. Let's define $\mu, \upsilon$ probability measures on some measure spaces $X, Y$ respectively, and $\pi$ a probability measure on the ...
Nerey's user avatar
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1 answer
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Example of finding the lower semi-continuous convex function whose subdifferential contains the support of an optimal transport

Related to this post, but slightly altered. Let $\mu$ uniform on $\{0\}\times [0,1]$, $\nu$ uniform on $\{-1,1\}\times [0,1]$. Consider the Kantorovich optimal transport problem of $\mu$ to $\nu$ with ...
J.R.'s user avatar
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3 votes
0 answers
64 views

Exercise 2.14 in Villani's "Topics in Optimal Transportation"

Exercise 2.14 in Villani's "Topics in Optimal Transportation" asks us to show the following: Let $Q=[0,1]^{n-1}$, and $\mu$ is uniform on $Q\times \{0\}$, $\nu$ uniform on $Q\times \{1\} \...
J.R.'s user avatar
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0 answers
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Trivial Optimal Couplings

I've been reviewing some basic optimal transport concepts (ref Peyré and Cuturi). I like the notation of the Kantorovich optimal transport problem. $$ L_C(a,b) = \min_{ P \in U(a,b) } \langle P , C \...
jeffery_the_wind's user avatar
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0 answers
32 views

Regularizing kernel and mollifier

I am learning the book "Optimal transport old and new" by Villani. We have the following notion of Regularizing kernels. My first question is: if this regularizing kernel is similar to ...
Jay's user avatar
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0 answers
29 views

Even regularizing kernel in $L_1$

I am learning the text book "Optimal transport for applied mathematician" by Filippo Santambrogio. In the Section 5.1, we try to prove that the Wasserstein distance satisfies the triangle ...
Jay's user avatar
  • 301
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0 answers
32 views

A condition for two cost functions to yield the same optimal transport cost?

Let $X,Y$ be two standard Borel spaces. Let $\mu,\nu$ be probability measures on $X$ and $Y$, and let $c_1,c_2 : X\times Y \to [0,1]$ be two measurable (but not necessarily lower semi-continuous) maps....
Guillaume Geoffroy's user avatar
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1 answer
37 views

A question on Monge formula- Optimal Transport

I have started reading optimal transport from the book "Optimal Transport for Applied Mathematicians" and I have a question regarding change of variables in Monge's formulation. We can ...
S_Alex's user avatar
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1 vote
0 answers
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What is the image of a transference plan?

Let $\mu, \nu$ be two positive Borel measures on $\mathbb{R}^d$ with the same mass. A probability measure $\pi$ on $\mathbb{R}^d\times\mathbb{R}^d$ is called a transference plan from $\mu$ to $\nu$ if:...
Ykiaz's user avatar
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1 vote
0 answers
51 views

transport equation interpretation

Premise: Theorem 5.34 from the book "Topics in Optimal Transportation" Let $X$ be $\mathbb{R}^n$. Let $(T_t)_{0\leq t\leq T_{*}}$ be a locally Lipschitz family of diffeomorphisms in $X$, ...
chintan's user avatar
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1 vote
0 answers
46 views

Convex combination of cyclically monotone sets

I want to show the following statement, but I am not sure how. Proposition(?): Let $C \in \mathbb{R}^d$ be a compact convex set, and let $u, v : C \to \mathbb{R}$ be smooth convex functions. Suppose $$...
Paruru's user avatar
  • 157
2 votes
0 answers
55 views

Bregman divergence from Wasserstein distance

I was wondering whether one has studied the Bregman divergence arising from a squared Wasserstein distance. More precisely, let $\Omega\subset \mathbb{R}^d$ be a compact set and $c\in \Omega\times \...
John's user avatar
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0 answers
43 views

Simplifying large scale nonlinear numerical optimization problem

I want to solve the optimization problem below numerically using GAMS. Given the formulation, I was wondering if there are suitable ways of transforming and or approximating the given function to ...
tjm's user avatar
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0 votes
0 answers
81 views

Distance between empirical distribution and population

Suppose we have $N$ points $z_1, \dots, z_N$. From this we can form the empirical distribution $\mu_z = \frac{1}{N} \sum_{i=1}^N \delta_{z_i}$. Now suppose we sample $N$ i.i.d. points from $\mu_z$, i....
900edges's user avatar
  • 2,039
0 votes
0 answers
28 views

Density of the maximum of 2 random variables with arbitrary dependence

Let $(X,Y)$ be a random vector taking values in $\mathbb{R}^2$ such that both $X$ and $Y$ are (marginally) distributed uniformly over the interval [0,1]. Let $Z= \max(X,Y)$ and $f$ denote the density ...
RMasini's user avatar
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1 vote
2 answers
155 views

Wasserstein Metric Inequality

This is the exercise: This exercise shows that “spreading out” probability measures makes them closer together. Define the convolution of a measure by: for any probability density function $\phi$, let ...
Raul Bataccs's user avatar
4 votes
1 answer
112 views

If $f > 0$, $\ker(L)$ contains only constant functions, where $L = - \Delta + \nabla (- \log(f)) \cdot \nabla$ (Villani, Subsec. 7.6)

In subsection(s) 7.5 (and 4.1) of Topics in Optimal Transportation, Cedric Villani states the following (I paraphrase): Take a probability measure $\mu \in \mathcal P_2(\mathbb R^n)$ with finite ...
ViktorStein's user avatar
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0 votes
0 answers
16 views

minimizing Earth Mover Distance

So I have a discretized magnitude spectrum $S \in \mathbb{R}^n$ ($n$ number of bins), and a set of frequencies $f_1, f_2, ..., f_m$ (not necessarily corresponding to any of the discretized bin ...
SmoothKen's user avatar
  • 429
1 vote
1 answer
194 views

Are linear interpolation curves on Wasserstein spaces absolutely continuous?

Let $\mathcal{P}_2$ the space of absolutely continuous probability measures on $\mathbb{R}^d$ with finite second moment equipped with the $2$-Wasserstein metric. Fix $\mu_0, \mu_1 \in \mathcal{P}_2.$ ...
Paruru's user avatar
  • 157
0 votes
0 answers
30 views

If $z=(x, v) \sim \pi(dx)\eta(dv)$, find $\eta(dv)$ such that $v'\sim\varpi$ where $z'=\psi(z)$ and $\psi$ invertible

I have a variable $z = (x, v)$ distributed according to $\lambda(dz) = \pi(dx)\eta(dv)$, where $\pi$ is fixed and known, but $\eta$ is unknown. I also have an invertible transformation $\psi$ and I ...
Euler_Salter's user avatar
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0 answers
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Questions on the book of Ambrosio-Gigli-Savaré <<Gradient flow>>

I am learning the book of Ambrosio-Gigli-Savaré "Gradient flow". However, I am blocked at the proof of proposition 8.5.2 (Optimal displacement maps are tangent). I don't understand where $\...
YJ_'s user avatar
  • 13
0 votes
0 answers
31 views

Equivalent condition on marginals

Let $\mu$ be a probability measure on $X$ (compact), $v$ a probability measure on $Y$ (compact), and let $\gamma$ be a probability measure on $X \times Y$: if $$\forall \psi \in C(X), \int_X \psi(x)d\...
nimaba99's user avatar
2 votes
0 answers
130 views

Distance between a distribution and an empirical distribution of samples

Looking for a reference on what I think is a standard result. For a measure $\mu \in P(\mathbb{R}^n)$, let $y_1, \dots, y_N \sim \mu$ be independent samples. The empirical distribution associated with ...
900edges's user avatar
  • 2,039
0 votes
0 answers
21 views

how to solve for wasserstein duality easily in a special case when 2-Wasserstein inequality constraint is binding

I was going through this nice paper ” A Simple and General Duality Proof for Wasserstein Distributionally Robust Optimization”, and one quick qu on applying Theorem 1 to my poject: What if in my ...
numpynp's user avatar
  • 61
1 vote
1 answer
95 views

Optimal Transport of radial measures

This is the exercise i). Find a transport map between $d\mu_d=\frac{1}{\pi}\mathbb{I}_{B_1(0)}dxdy$ and $d\mu_c=\frac{1}{8\pi}\left(4-|x|^2\right)_+dxdy$. ii) Compute $C^K_{\rho}(\mu_d,\mu_c)$ where ...
Raul Bataccs's user avatar
0 votes
0 answers
108 views

Solve $y=\operatorname{erf}(x+c)+\operatorname{erf}(x-c)$ for $x$

$\newcommand{\erf}{\operatorname{erf}}\newcommand{\erfc}{\operatorname{erfc}}$ Is there a closed form solution for $y=\erf(x+c)+\erf(x-c)$? More specifically, I want to solve $$\erf(\frac{y}{\sqrt{2}})...
Hyperplane's user avatar
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47 views

What is the conventional definition of $d(x,y)^{p}$?

In Optimal transportation, and more precisely in "Optimal Transport: Old and New" (Definition. 6.1, on page 106 - actually on page 111 out of 998, in this link), the Wasserstein distance $...
Ommo's user avatar
  • 349
0 votes
1 answer
144 views

Optimal Transport Theory: Find a sequence of functions that converge weakly to 1/2 [closed]

I'm trying solve the following problem: Find a sequence $T_n$ of transport maps from $\mathbb{I}_{[0,1]}dx$ to $\mathbb{I}_{[0,1]}dx$ such that $T_n$ converges weakly to $\frac{1}{2}$. Deduce that the ...
Raul Bataccs's user avatar

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