Questions tagged [optimal-transportation]
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260 questions
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Exchanging the integral and infimum on the space of couplings
Let $\mu,\nu$ be probability measures on $\mathbb{R}^d$ with finite $p$-th moment ($p\in [1,\infty)$) and define the set of couplings by $\mathcal{C}(\mu,\nu)$ i.e. the set of probability measures on ...
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49
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Transport map to lower dimension?
Let $S^{d-1}$ be the sphere in $\mathbb{R}^d$.
Given a $C^\infty$ function $f \colon S^{d-1} \to \mathbb{R}$, define $g \colon S^{d-1} \to S^{d-1}$ as $g(x) = \exp_x(\nabla f(x))$, where $\nabla f(x)$ ...
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Should coffee machines be placed at the region's boundary?
This is a continuation of Should coffee machines be deconcentrated?
Recall that some region is denoted by convex and compact $E\subset \mathbb R^2$. $N\ge 1$ coffee machines are provided for the ...
- 1,409
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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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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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1
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137
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Wasserstein distance and put function
Let $\mathcal P$ be the set of probability distributions on $\mathbb R$ of finite first order, i.e. $\mu\in\mathcal P$ if
$$\int_{\mathbb R} |t|\mu(dt)<\infty.$$
For $\mu\in\mathcal P$, define its ...
- 1,409
2
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1
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141
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(Sub)Optimality of random transport
Problem Setup: Consider the intervals $I_R = [a_R, b_R]$ and $I_B = [a_B,b_B]$. Let $F_R$ and $F_B$ denote the CDF of distributions with support on the intervals $I_R$ and $I_B$. I draw $k$ red and ...
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1
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129
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Wasserstein distance of push-forward measures
I asked this same question on MSE, but with no luck, so I am trying to ask here.
Consider two measures $\mu , \nu$ on $\mathbb{R}^n$. Now consider a map (a priori only measurable, but feel free to add ...
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Should coffee machines be deconcentrated?
We model some region by convex and compact $E\subset \mathbb R^2$. $N\ge 1$ coffee machines are provided for the people living on $E$, of capacities $\alpha_1,\ldots, \alpha_N>0$. Assume the ...
- 1,409
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Does complete and separable Wasserstein space imply a complete base space?
Also asked on math.SE.
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 ...
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Existence of first variation
I am trying to compute the first variation of the functional
$$\mathcal F(\rho) = \int_{\Omega} R(x;\rho) d\rho(x)$$
where $R$ is some function of $x$ that also depends on $\rho$. Here $\rho$ is a ...
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471
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Proving the inequality involving Hausdorff distance and Wasserstein infinity distance
Prove the inequality
$$d_{H}(\mathrm{spt}(\mu),\mathrm{spt}(\nu))\leq W_{\infty}(\mu,\nu)$$
where $d_H$ denotes the Hausdorff distance between the supports of the measures $\mu$ and $\nu$, and $W_\...
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106
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When is there a Lipschitz Kantorovich Potential?
Let $c:\mathbb{R}^d\times \mathbb{R}^d\to [0,\infty)$ be a Lipschitz cost function and consider the optimal transport problem
$$
C(\mu,\nu):=\inf_{\pi}\, \int c(x,y)\,\pi(dxdy)
$$
where, as usual, the ...
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1
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Does this maximisation problem admit a finite upper bound?
Let $\mathcal M_2$ be the space of real $2\times 2$ matrices and $\mathcal S_2\subset \mathcal M_2$ be its subset consisting of positive semidefinite elements, i.e. $A\in \mathcal S_2$ iff $A$ is ...
- 1,409
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Is the following set compact w.r.t. the Wasserstein distance?
Fix a finite first moment probability measure $q\in\mathcal{P}_1(\mathbb R ^d)$, and real numbers $K,M,R$. Consider the following set:
$$A:=\left\{p\in\mathcal{P}_1(\mathbb R ^d): \int |x|dp\leq K, \...
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151
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Is this set $\sigma$-compact in the Wasserstein space?
This is a follow-up to this question.
Fix a finite first moment probability measure $q\in\mathcal{P}_1(\mathbb R ^d)$, and real numbers $K,M,R$. Consider the following set:
$$A:=\left\{p\in\mathcal{P}...
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1
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1
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118
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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}}
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\newcommand{\bP}{\mathbb{P}}
\newcommand{\bE}{\mathbb{E}}
\newcommand{\bF}{\mathbb{F}}
\newcommand{\bD}{\mathbb{...
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Connection between Wassertein-2 metric and difference in variance
Given two probability densities $\mu\in\mathcal P(\mathbb R^d)$ and $\nu\in\mathcal P(\mathbb R^d)$, we define their Wasserstein-$p$ metric as
$$
W_p^p(\mu, \nu)=\inf_{\gamma\in \Gamma(\mu, \nu)}\int_{...
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Sequential compactness of a sequence of curves of Borel probability measures
$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\DeclareMathOperator*{\supp}{supp}
\newcommand{\bR}{\mathbb{R}}
\newcommand{\bT}{\mathbb{T}}
\newcommand{\bN}{\mathbb{N}}
\newcommand{\bP}{\mathbb{...
- 825
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Weak convergence + convergence of the norm implies strong convergence in Orlicz spaces
It is known [1, proposition 3.32] and a classical trick in PDEs that, in any uniformly convex Banach space $X$, weak convergence $x_n\rightharpoonup x$ together with convergence of the norm $\|x_n\|_X\...
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231
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Optimal transport plan induced by an optimal transport map
I am reading Santambrogio's book on optimal transport, remark 1.19. Let's consider an optimal transport problem between $(X,\mu)$ and $(Y,\nu)$.
(Remark 1.19) ... every time that we know that any ...
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1
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1
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127
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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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0
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54
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How to deal with minimizing a flat objective function
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 ...
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1
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1
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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 ...
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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}$$ on ...
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39
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Comonotone solution for Optimal Transport problems with supermodular surplus
In Alfred Galichon's book Optimal Transport Methods in Economics the foollowing result is stated for OT problems on the real line.
Theorem 4.3.(i) Assume that $\Phi$ is supermodular. Then the primal ...
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2
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485
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How to get this inequality in Santambrogio's book about optimal transport?
Let $\hat{\varrho}, \tilde{\varrho}$ be probability density functions on $\mathbb R^d$ where $\tilde{\varrho} \in L^{\infty} (\mathbb R^d)$. For $\varepsilon \in [0, 1]$, we define $\varrho_{\...
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Wasserstein bounds of interpolation measures
Assume we are given two densities, $p_0$ and $p_1$ on $\mathbb{R}^d$, and define (up to the normalization constant) the interpolation $p_t \propto p_0^{1-t} p_1^t$, which interpolates between $p_0$ ...
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1
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1
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Lipschitz approximation of a probability measure with finite $1$-st moment by the ones with finite $p$-th moment
For $p \in [1, \infty)$, let $\mathcal P_p (\mathbb{R^d})$ be the space of Borel probability measures on $\mathbb R^d$ with finite $p$-th moment. We endow $\mathcal P_p (\mathbb{R^d})$ with the ...
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Any useful bases for the topology induced by the $t$-Wasserstein distance?
I am working on $\mathbb R ^d$ equipped with the usual Euclidean metric. I know of one nice base for $\mathcal W _t$, namely: $$\left\{ B_p (r) : r>0, p=\sum_{i=1} ^n \alpha_i \delta_{x_i},\text{ ...
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1
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Gradient flows: evolution of geodesics
I’m trying to understand if, when I move the marginals of a Wasserstein geodesic along a contractive flow, the geodesic between the new probability measures is “near” to the geodesic
connecting the ...
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385
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Continuity equation $\partial_t \mu_t+\operatorname{div} (v_t \mu_t)=0$: are these two notions of weak solution equivalent?
Let $\Omega$ be an open connected convex subset of $\mathbb R^d$. Let $\mathcal P (\Omega)$ be the space of Borel probability measures on $\Omega$. Let $C_0 (\Omega)$ be the space of real-valued ...
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Perturbation of Wasserstein distance: looking for references
I am doing readings related to Optimal transport which is new to me and I often encounter the following statement regarding a sort of derivative of the Wasserstein distance: $u$ and $v$ be two ...
- 1,101
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2
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1k
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Wasserstein distance between product measures
For two probability measures $\mu,\nu$ on $\mathbb{R}^n$, let
$$W_p(\mu,\nu)^p=\inf_{\pi}\int\|x-y\|^p_2\ \pi(dx\ dy)$$
denote the $p^\text{th}$ Wasserstein distance between $\mu, \nu$, where the ...
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Are 1-Wasserstein and 2-Wasserstein distances between multivariate normal distributions equivalent?
The $p$-Wasserstein between two measures $\nu_1$ and $\nu_2$ on $X$ is given by
$$W^p_p(\nu_{1},\nu_{2})=\underset{\pi\in\Gamma(\nu_{1},\nu_{2})}{\inf}\int_{\mathbf{\mathcal{X}}^{2}}d(x,y)^p\pi(dx,dy)$...
- 1,058
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59
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Random solutions to non-Lipschitz ODEs, optimal transport, and general solutions to the continuity equation
I am reading Cedric Villani’s book “Optimal Transport: old and new” and I am stuck on one paragraph (see page 26/27 in this book). He speaks about random solutions to an ODE and I simply cannot figure ...
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Some stability and estimate of the optimal transport map (Brenier map)
Let $\mu$ and $\nu$ be two probability measures with finite moments (in $\mathcal{P}_2(\mathbb{R})$) equipped with 2-Wasserstein distance. Let $F_\mu$, $F_\nu$ be their cumulative distribution ...
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1
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1
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262
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What does $g_x^{-1}$ mean where $g$ is a Riemannian metric?
I am reading this paper about the Wasserstein-Fisher-Rao distance and they define the Wasserstein-Fisher-Rao distance on a manifold as follows (page 9):
Here, $M$ is a compact Riemannian manifold, $\...
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2
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256
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Are the sublevel sets of Boltzmann entropy compact in Wasserstein distance?
For Lebesgue-absolutely continuous probability measures $\rho\ll \mathcal{L}^d$ in the whole space $\mathbb{R}^d$ with finite second moments (i-e $\rho\in \mathcal{P}^2_{ac}(\mathbb{R}^d)$), let
$$
\...
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Let $\mu : [0, T] \to \mathcal P_2^a (\mathbb R^d), t \mapsto \mu_t$ be absolutely continuous. Is $t \mapsto \mathcal H (\mu_t)$ continuous?
We endow the space $\mathcal P_2^a (\mathbb R^d)$ of absolutely continuous probability measures with finite second moment with the Wasserstein distance $W_2$. Let $\mathcal H (\mu)$ be the relative ...
- 825
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1
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210
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Some continuity issues of the optimal transport map (Brenier map)
Let $\mu$ and $\nu$ be two probability measures with finite moments (in $\mathcal{P}_2(\mathbb{R})$) equipped with 2-Wasserstein distance. Let $F_\mu$, $F_\nu$ be their cumulative distribution ...
- 54
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Can we control the Wasserstein metric between $\mu$ and $\nu$ by their moment difference?
Fix $p \in [1, \infty)$. Let $(\mathcal P_p(\mathbb R^d), W_p)$ be the Wasserstein space of all Borel probability measures on $\mathbb R^d$ with finite $p$-th moment. Let $D_p$ be the collection of ...
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87
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Instances of c-concavity outside of optimal transport?
Let $X$ and $Y$ be metric spaces, and let $c:X\times Y\rightarrow \mathbb{R}$ be a nonnegative function which we refer to as a cost. For any $\phi:X\rightarrow \mathbb{R}$ and $\psi:Y\rightarrow \...
- 232
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0
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137
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On the continuity with respect to the increasing convex order
For $p\ge 1$, let $\mathcal P_p(\mathbb R)$ be the set of probability measures on $\mathbb R$ of finite $p^{\rm th}$ moment. Denote by $W_p$ the Wasserstein metric of order $p$ and by $\preceq$ the ...
- 1,409
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148
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Exponential map for tangent space of space of distributions $\mathscr{P}_2(X)$
In Chapter 8 of the book Gradient Flows In Metric Spaces and in the Space of Probability Measures by Ambrosio et al., the tangent space to the space of distributions on $X$ (let's say $X=\mathbb{R}^d$)...
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1
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310
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Local Lipschitzness of parameterization of Gaussians in Wasserstein space
Fix a positive integer $n$ and consider the $2$-Wasserstein space $\mathcal{P}_2(\mathbb{R}^n)$. Let $X$ be the cone of $n\times n$ symmetric positive semidefinite matrices with Frobenius norm and ...
- 169
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1
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54
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How is this interpolating curve well-defined in the minimizing movement scheme?
Let $\Omega$ be a compact domain of $\mathbb R^d$. Let $\mathcal P (\Omega)$ be the space of probability measures on $\Omega$. For each $\tau >0$, let $(\varrho^\tau_{(k)})_{k \in \mathbb N} \...
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1
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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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0
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66
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Continuous trajectory of midpoints of length-minimizing geodesics
Let $M$ be a smooth Riemannian manifold, $x$ be a point in $M$, and $\lambda:[0,1]\to M$ be a continuous path. Can we find a family of length-minimizing constant speed geodesics $\gamma_t:[0,1]\to M$ ...
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3
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0
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66
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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
$$...
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