Questions tagged [optimal-transportation]
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260 questions
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$\sigma$-compactness of probability measures under a refined topology
Denote Polish spaces $(X, \tau_x)$ and $(Y, \tau_y)$, where $X$ and $Y$ are closed subsets of $\mathbb{R}$. Consider a Borel measurable function $f: (X \times Y, \tau_x \times \tau_y) \rightarrow \...
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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 ...
- 956
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65
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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},(\...
- 169
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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 \...
- 825
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131
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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 ...
- 6,853
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82
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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 ...
- 303
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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 ...
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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-...
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101
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$CD(K,N)$ 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 ...
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328
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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 ...
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146
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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 ...
- 323
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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}(...
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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 ...
- 345
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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 - ...
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a generalization of Monge-Kantorovich Problem
I am thinking about the martingale version of Monge-Kantorovich Problem.
Let $\mu(x)$ and $\nu(y)$ denote two density laws on $\mathbb{R}$, and define $M(\mu,\nu)$ the set of densities $f(x,y)$ on $\...
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889
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Simplify Wasserstein distance between Gaussians with binary cost function
Let $\mu_1$ and $\mu_2$ be 1D gaussian distributions with means $m_1$ and $m_2$ respectively and common variance $\sigma$. Let $\Omega$ be a closed subset of $\mathbb R^2$, and consider the cost ...
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307
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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(...
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Is there $\varepsilon \in (0, 1)$ such that $\sup_{t \in [0, \varepsilon]} [\ell_t]_\beta < \infty$?
$
\newcommand{\bR}{\mathbb{R}}
\newcommand{\bT}{\mathbb{T}}
\newcommand{\bN}{\mathbb{N}}
\newcommand{\bP}{\mathbb{P}}
\newcommand{\bE}{\mathbb{E}}
\newcommand{\bF}{\mathbb{F}}
\newcommand{\bD}{\mathbb{...
- 825
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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 \...
- 5,405
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270
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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 ...
- 660
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685
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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 ...
- 1,269
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108
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Iterated optimal transport
Suppose we are interested in two consecutive transport plans (in the Kantorovich formulation). That is, we are given finite sets $X$, $Y$ and $Z$, endowed with probability measures $\mu_X$, $\mu_Y$ ...
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If $\pi$ is a coupling between $f_*\mu_X, g_*\mu_Y$ and $\pi = (f(x),g(y))_* \sigma$, then $\sigma$ is a coupling between $\mu_X,\mu_Y$
I am trying to read this paper: "The Gromov-Wasserstein distance between networks and stable network invariants" https://arxiv.org/abs/1808.04337. In this paper, they have the following ...
- 305
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86
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Is it true that $\xi \in \partial G (v)$ implies $\frac{\xi}{F'(\phi (v))} \in \partial \phi (v)$?
I am reading the introduction of Chapter 10 in the book Gradient Flows by Ambrosio and his coauthors.
As we have seen in Section 1.4, in the classical theory of subdifferential calculus for proper, ...
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262
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What does $g_x^{-1}$ mean where $g$ is a Riemannian metric?
I am reading this paper about the Wasserstein-Fisher-Rao distance and they define the Wasserstein-Fisher-Rao distance on a manifold as follows (page 9):
Here, $M$ is a compact Riemannian manifold, $\...
- 305
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313
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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 ...
- 825
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195
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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 ...
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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}$, ...
- 547
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243
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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,...
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283
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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 $...
- 5,405
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$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 ...
- 5,405
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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 ...
- 5,405
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242
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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
$$
P_1(X,d)\triangleq \left\{
\nu:\,(\exists x_0\in X)\, \int_{x\in X} d(x,x_0)d\...
- 5,405
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Metric 1-current decomposition
I've been reading Paolini-Stepanov arcticle and in section 4, at page 6, they define a metric current from a transport:
$$T_{\eta}(\omega)=\int_{\Theta}[[\theta]](\omega)d\eta(\theta),$$
which ...
- 391
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129
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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 ...
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60
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Optimal transport for sum of two costs
Let $X$ be a finite set and $\sigma_0$, $\sigma_1$ two fixed measures on $X$ with $\sigma_0(X)=\sigma_1(X)$. A transportation plan is a measure $\mu$ on $X\times X$ whose projections on the first and ...
- 1,074
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Lipschitz approximation of a probability measure with finite $1$-st moment by the ones with finite $p$-th moment
For $p \in [1, \infty)$, let $\mathcal P_p (\mathbb{R^d})$ be the space of Borel probability measures on $\mathbb R^d$ with finite $p$-th moment. We endow $\mathcal P_p (\mathbb{R^d})$ with the ...
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Generalised Lebesgue transform continuous wrt. strict topology?
Let $X$ and $Y$ be $\sigma$-compact spaces, and $\mu$ [resp. $\nu$] be a regular Borel probability measure on $X$ [resp. $Y$].
For a bounded continuous $c:X\times Y\rightarrow\mathbb{R}$, consider the ...
- 463
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354
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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 $\...
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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 ...
- 825
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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\, \...
- 5,405
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64
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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 ...
- 553
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261
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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 $\theta\mapsto \mathbb{P}_{\theta}$ ...
- 5,405
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1k
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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 ...
- 1,127
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1
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124
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Finding $P$ points among $N$ to approximate a probability density function?
Let $f$ be a probability density function (positive such that $\int_{\mathbb{R}} f(x) \mathrm{d} x = 1$) and $X_0 = \{x_n\}_{1\leq n \leq N}$ be $N$ given real points. We also fix $1 \leq P \leq N$ ...
- 2,306
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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
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169
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Optimal transport-like problem where the objective depends on conditional probability distribution
$\DeclareMathOperator\marg{marg}$I would like to know if the following problem can be studied as an optimal transport problem, possibly imposing additional assumptions on the data.
Consider two sets $\...
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1
answer
524
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Optimal transport between two matrices
I'm investigating the use of optimal transport to define a distance between non-negative matrices that satisfy the condition $\mathbf e^\top\mathbf M\mathbf e = 1$ (i.e., the sum of all elements in ...
- 243
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433
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Derivative of Wasserstein distance $W^p_p$ along solutions of the continuity equation (contradicting statements in different sources)
Let $(\rho^{(i)}_t,{\bf v}^{(i)}_t)$ for $i = 1,2$ be two solutions of the continuity equation $$\partial \rho^{(i)}_t + \nabla\cdot \left({\bf v}^{(i)}_t \rho^{(i)}_t\right) = 0 \label{1}\tag{1}$$ on ...
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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 (...
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