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Schrödinger Bridge for other costs

Stochastic control formulations of the Schrödinger bridge problem between $\mu,\nu$ are well known (e.g Chen et al Eq. 4.23) $$\inf \limits_{p_t, v_t} \int_0^T \int \frac{1}{2}\lvert v_t\rvert^2 p_t ...
nico's user avatar
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2 votes
1 answer
60 views

A variant of (discrete) optimal transport problem

Let $\alpha=(\alpha_1,\ldots,\alpha_m)\subset\mathbb R^m_+$ and $\beta=(\beta_1,\ldots,\beta_n)\subset\mathbb R^n_+$ be given and satisfy $$\sum_{i=1}^m \alpha_i =1 = \sum_{j=1}^n\beta_j.$$ Define $\...
Fawen90's user avatar
  • 437
1 vote
0 answers
65 views

Representation formula for the continuity equation on a separable Hilbert space

The following is an informal question for which I'd like to (ideally) find a reference. I'm quite a novice in this area but would be happy to find a reference to a theorem along the following lines (...
Gregor Samsa's user avatar
2 votes
0 answers
64 views

Approximating solutions to Monge-Ampere from optimal transport plans

I am interested in finding numerical solutions to a Monge-Ampere type equation for applications in physics. Due to the close connection between Monge-Ampere and optimal transport and the well ...
Yly's user avatar
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3 votes
2 answers
150 views

Getting Wasserstein closeness from a derivative estimate

In my setting, $\mu$ and $\nu$ are probability measures on $\mathbb{R}^{2}$ with compact support. For any function $f\in{C^{2}_{b}(\mathbb{R}^{2})}$, I have the estimate: $$ |\mathbb{E}_{\mu}(f)-\...
David Pechersky's user avatar
3 votes
1 answer
102 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 ...
Analyst's user avatar
  • 475
1 vote
1 answer
79 views

Is the Wasserstein distance to the empirical measure minimized by the underlying distribution?

Let $S$ be a metric space and denote the set of probability measures on $S$ by $\mathcal{P}(S)$. Fix $\mu\in \mathcal{P}(S)$ and denote the law of $N\geq 1$ i.i.d samples $X=(X_1,\ldots,X_N)$ from $\...
joemrt's user avatar
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2 votes
0 answers
49 views

Riemannian submanifolds of $2$-Wasserstein space

In the article "Wasserstein Geometry Of Gaussian Measures" by Asuka Takatsu the author shows how the space of d-dimensional Gaussian probability measures with non-singular covariance ...
Annie's user avatar
  • 71
8 votes
1 answer
261 views

Mass transportation proof of the Gaussian isoperimetric inequality?

In his book "Topics in optimal transportation", Graduate Studies in Mathematics 58, AMS 2003, Villani presents a proof, due to Gromov, of the classical isoperimetric inequality in Euclidean ...
Xiazhong Zhu's user avatar
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Wasserstein compactness of sublevel sets of relative entropy

Let $\mathcal{P}({\mathbb{R}^d})$ denote the set of Borel probability measures on $\mathbb{R}^d$, and let $\pi \in \mathcal{P} (\mathbb{R}^d)$. It is known that the sets $\{ \mu \in \mathcal{P}(\...
pseudocydonia's user avatar
1 vote
0 answers
156 views

$L^2$ metric on $\textrm{Diff}(M)$ and geodesics

The paper Geometry of diffeomorphism groups, complete integrability and optimal transport mentions the following: The group $\textrm{Diff}(M)$ carries a natural $L^2$-metric $\displaystyle \langle\...
Kaira's user avatar
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2 votes
1 answer
148 views

For diffeomorphism $f$, if $X$ and $f(X)$ are both Gaussian, then $f$ is affine

I am trying to prove the following. Let $f:\mathbb{R}^{n}\to \mathbb{R}^n$ be a diffeomorphism. If $X$ and $f(X)$ are both $n$ -dimensional Gaussian variables, then $f$ is affine. That is, there ...
Kaira's user avatar
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0 answers
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Lipschitzness of conditional law of a stochastic filtering problem wrt the Wasserstein distance

Let $(X_t)_{t\ge 0}$ and $(Y_t)_{t\ge 0}$ be a pair of stochastic processes taking values in $\mathbb{R}^n$ and in $\mathbb{R}^m$; defined on a filtered probability spaces $(\Omega,\mathcal{F},(\...
Justin_other_PhD's user avatar
1 vote
1 answer
71 views

Strict convexity of the cost function is enough to ensure the existence and uniqueness of the optimal transport map

Let $X=Y = \mathbb R^d$ and $c:X \times Y \to [0, \infty)$ be Borel measurable. Let $\mu, \nu$ be Borel probability measures on $X,Y$ respectively. Let $\mathcal T$ be the set of all Borel measurable ...
Akira's user avatar
  • 713
4 votes
1 answer
247 views

Optimal Transport: how is this transport map Borel measurable?

I'm reading Theorem 1.17. and its proof at page 14 of Santambrogio's Optimal transport for applied mathematicians. The content is not hard but a little bit long (because of related detail). Please ...
Akira's user avatar
  • 713
1 vote
1 answer
104 views

Bounds on the Bures–Wasserstein distance

The Bures–Wasserstein distance between $n\times n$ positive semidefinite matrices $A$ and $H$ is defined to be $$ d(A,H) := \left[ \operatorname{tr} A + \operatorname{tr} H - 2\operatorname{tr} (A^{1/...
eepperly16's user avatar
0 votes
0 answers
41 views

Even transport map and Brenier map in optimal transport?

I am working on a topic related to even transport map, i.e. $T(-x) = T(x)$. It's known that if $T\# \mu =\nu$ where $\mu = f(x)dx$ and $\nu = g(y)dy$, then $T$ satisfies the Jacobian equation $g(T(x)) ...
Dongwei's user avatar
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1 vote
1 answer
54 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 ...
Akira's user avatar
  • 713
1 vote
1 answer
118 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 ...
Akira's user avatar
  • 713
2 votes
0 answers
113 views

Optimal transport: how is the use of disintegration theorem valid in this construction of $\widetilde{\phi}$?

Let $X,Y$ be Polish spaces and $\mathcal P(X)$ the space of all Borel probability measures on $X$. Fix $\mu\in \mathcal P(X), \nu \in \mathcal P(Y)$. Let $\pi \in \Pi(\mu, \nu)$, i.e., $\pi \in \...
Akira's user avatar
  • 713
1 vote
1 answer
112 views

Optimal transport: how $\varphi^c$ can be written as $\varphi^c = \lim _{\ell \rightarrow \infty} \psi_{\ell}$?

Let $X,Y$ be Polish spaces and $c:X \times Y \to [0, \infty]$ lower semi-continuous. There is a sequence $(c_\ell)_{\ell \in \mathbb N}$ with $c_\ell:X \times Y \to [0, \infty)$ of bounded Lipschitz ...
Akira's user avatar
  • 713
1 vote
0 answers
43 views

$c$-cyclical monotonicity: does this proof hold if $f \equiv +\infty$ or $\int c \mathrm d \gamma = +\infty$?

I'm reading the proof of Theorem 1.38. from section 1.6.2 $c$-cyclical monotonicity and duality of Santambrogio's Optimal transport for applied mathematicians. My understanding: It seems for the ...
Akira's user avatar
  • 713
1 vote
1 answer
109 views

Are convex functions on manifolds the same as $c$-convex functions, where $c(x,y)=d(x,y)^2/2$?

I am reading the following book on optimal transport. While reading I came across the following definition of $c-$convexity. Given $X$ and $Y$ metric spaces, $c: X \times Y \rightarrow \mathbb{R}$, ...
Student's user avatar
  • 591
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0 answers
42 views

"More" cyclical monotonicity

Let $X$ and $Y$ be some finite sets. For a given function $f:X\times Y\rightarrow \mathbb{R}$, we say a set $S\subset X\times Y$ is $f$-cyclically monotone if for any sequence $(x_1,y_1),...,(x_n,y_n)\...
user_XL's user avatar
1 vote
1 answer
84 views

Continuity equation for a density of a measure

From the paper of Ambrosio-Crippa, it is known that if $\beta:\mathbb R^d\times[0, T[\longrightarrow\mathbb R^d$ is suitably regular, then the system $$ \begin{cases} \dfrac{\partial\mu}{\partial t}(x,...
Redeldio's user avatar
  • 167
2 votes
0 answers
69 views

Continuity of the traffic intensity

Note: We say that a Borel measure on $\mathbb R^n$ has a continuous density if it is absolutely continuous with respect to Lebesgue measure and has a continuous Radon Nikodym derivative. Let $\mu$ be ...
Nate River's user avatar
  • 2,574
1 vote
0 answers
56 views

Conditions for existence of an optimal transport map between finitely supported probability measures

Let $E$ be a polish and let $P$ and $Q$ be finitely supported probability measures on $X$. What conditions are required to ensure that: for every $\delta>0$ there exists a $\delta$-optimal ...
James_Gromov's user avatar
1 vote
1 answer
76 views

Distance between empirical measures and thickened version

Let $\mathcal{H}$ be a separable Hilbert space and let $x_1,...,x_n$ be points in $\mathcal{H}$. Let $\varepsilon >0 $ be given and consider the measures $$ \mu := \frac1{n}\,\sum_{i=1}^n\, \...
ABIM's user avatar
  • 4,881
1 vote
1 answer
84 views

Approximation of two densities with a single transformation

Let $p_1$ and $p_2$ be two probability densities and $X_i\sim N(\mu_i,\Sigma_i)$. Write $w(X)\sim p$ if the law of the random variable $w(X)$ has a density equal to $p$. For general densities $p_i$, ...
jack412's user avatar
  • 63
0 votes
0 answers
62 views

solution of equivalent problem Kantorovich for case squared distance function

We know that the Kantorovich duality when the cost function is the square Euclidean distance is equivalent to $$ \inf_{(\tilde\varphi,\tilde\psi)\in \tilde\Phi_c} J(\tilde\varphi,\tilde\psi) = \sup_{\...
Giovanni Arquimedes Wences Naj's user avatar
7 votes
0 answers
189 views

What are the "applications" of quantum optimal transport?

A quantum version of the Monge-Kantorovich optimal transport problem aims at optimizing a Hermitian cost matrix $C$ over the set of all bipartite coupling states $\rho_{AB}$, s.t. both of its reduced ...
user avatar
2 votes
1 answer
190 views

Optimal transport for applied mathematicians: how does $\varphi (x) = \inf_{y \in Y} [c(x, y) - \psi (y)] \neq -\infty$ follow in Theorem 1.37?

I'm reading a proof of Theorem 1.37 from Santambrogio's Optimal transport for applied mathematicians: calculus of variations, PDEs, and modeling. First, I quote related definitions. Let $X,Y$ be ...
Analyst's user avatar
  • 475
0 votes
2 answers
148 views

How to compute the unique disintegration w.r.t. the first coordinate?

Set $\pi=\frac{1}{4}(\delta_{(1,0)}++\delta_{(1,3)}+\delta_{(1,1)}+\delta_{(2,2)})$. Suppose that $\pi\in\Pi(\mu,\nu)$. How to get the disintegration of $\pi$ with respect to $\mu$?
Hermi's user avatar
  • 216
2 votes
0 answers
64 views

Does $\mathscr{H}^{d-1} (A)<+\infty$ for $A\subset \mathbb R^d$ imply $A$ is (Borel) measurable?

I'm reading section 1.3.1 The quadratic case in $\mathbb{R}^{d}$ at page 17 from Santambrogio's Optimal Transport for Applied Mathematicians. The PDF is freely available from here. Let $\mu$ be a ...
Akira's user avatar
  • 713
5 votes
1 answer
307 views

Uniqueness of Kantorovich potentials?

$\newcommand{\R}{\mathbb R}$Take $\Omega\subset \R^d$ bounded, convex, and smooth. Consider the optimal transport problem with cost $c(x,y)=\lvert x-y\rvert^2$, leading to the quadratic Wassersein ...
leo monsaingeon's user avatar
1 vote
0 answers
34 views

Transport-type duality for preduals of $C^{k,1}$-functions

Let $\Omega$ be a non-empty, simply connected, and open subset of $\mathbb{R}^d$ for some positive integer $d$. Let $k$ be a non-negative integer. Consider the Banach space $C^{k,1}_0(\Omega)$ ...
ABIM's user avatar
  • 4,881
2 votes
0 answers
104 views

Eigenvalues of Witten Laplacian induced by log-concave probability measure on manifold

Let $M$ be a closed $n$-dimensional Riemannian manifold and let $\mu=e^{-V}d\mathrm{vol}_M$ be a log-concave probability measure on $M$, such that the pair $(M,\mu)$ verifies the so-called Bakry-Emery ...
dohmatob's user avatar
  • 6,338
1 vote
0 answers
135 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 ...
Akira's user avatar
  • 713
0 votes
1 answer
74 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 ...
Akira's user avatar
  • 713
1 vote
0 answers
124 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\...
ABIM's user avatar
  • 4,881
4 votes
1 answer
277 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} \...
leo monsaingeon's user avatar
3 votes
0 answers
222 views

Radon-Nikodym derivative of vector-valued measure with respect to another vector-valued measure

Let $(X, | \cdot |)$ be a Banach space. I am interested in whether one can extend the definition of the Kullback-Leibler divergence $$ \text{KL}(\mu \ \Vert \ \nu) := \int_{\Omega} \ln\left(\frac{\...
ViktorStein's user avatar
2 votes
1 answer
131 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 ...
dohmatob's user avatar
  • 6,338
7 votes
3 answers
298 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 ...
Carlos_Petterson's user avatar
1 vote
0 answers
60 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 ...
dohmatob's user avatar
  • 6,338
1 vote
1 answer
51 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 ...
avk255's user avatar
  • 533
0 votes
1 answer
199 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}...
ECL's user avatar
  • 241
1 vote
0 answers
32 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(...
Silentmovie's user avatar
0 votes
2 answers
185 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 $...
ABIM's user avatar
  • 4,881
2 votes
0 answers
75 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 ...
Jun's user avatar
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