Questions tagged [optimal-transportation]
The optimal-transportation tag has no usage guidance.
260 questions
3
votes
0
answers
49
views
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 $\nabla f(x)$ ...
- 171
0
votes
1
answer
89
views
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 ...
- 305
1
vote
1
answer
60
views
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
1
vote
1
answer
108
views
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 $\mu_X$, $\mu_Y$ ...
- 13
3
votes
2
answers
614
views
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
2
votes
1
answer
141
views
(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 ...
- 25
2
votes
1
answer
129
views
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 ...
- 749
2
votes
1
answer
137
views
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
0
votes
0
answers
52
views
Path-homotopy in Wasserstein space
Consider two vector fields $b_0,b_1\in C^2([0,1]\times\mathbb{R}^d;\mathbb{R}^d)$ and the solutions $\rho_0,\rho_1\in AC([0,1];\mathcal{P}_2(\mathbb{R}^d))$ to the associated Fokker-Planck equations
$$...
- 33
2
votes
0
answers
106
views
When is there a Lipschitz Kantorovich Potential?
Let $c:\mathbb{R}^d\times \mathbb{R}^d\to [0,\infty)$ be a Lipschitz cost function and consider the optimal transport problem
$$
C(\mu,\nu):=\inf_{\pi}\, \int c(x,y)\,\pi(dxdy)
$$
where, as usual, the ...
2
votes
1
answer
142
views
Existence of first variation
I am trying to compute the first variation of the functional
$$\mathcal F(\rho) = \int_{\Omega} R(x;\rho) d\rho(x)$$
where $R$ is some function of $x$ that also depends on $\rho$. Here $\rho$ is a ...
- 21
0
votes
0
answers
54
views
How to deal with minimizing a flat objective function
Problematic (Debiased Sinkhorn barycenter, proposed by H.Janti et al.): Let $\alpha_1, \ldots, \alpha_K \in \Delta_n$ and $\mathbf{K}=e^{-\frac{\mathrm{C}}{\varepsilon}}$. Let $\pi$ denote a sequence ...
- 121
7
votes
1
answer
737
views
Should coffee machines be deconcentrated?
We model some region by convex and compact $E\subset \mathbb R^2$. $N\ge 1$ coffee machines are provided for the people living on $E$, of capacities $\alpha_1,\ldots, \alpha_N>0$. Assume the ...
- 1,409
0
votes
0
answers
39
views
Comonotone solution for Optimal Transport problems with supermodular surplus
In Alfred Galichon's book Optimal Transport Methods in Economics the foollowing result is stated for OT problems on the real line.
Theorem 4.3.(i) Assume that $\Phi$ is supermodular. Then the primal ...
2
votes
1
answer
73
views
Connection between Wassertein-2 metric and difference in variance
Given two probability densities $\mu\in\mathcal P(\mathbb R^d)$ and $\nu\in\mathcal P(\mathbb R^d)$, we define their Wasserstein-$p$ metric as
$$
W_p^p(\mu, \nu)=\inf_{\gamma\in \Gamma(\mu, \nu)}\int_{...
- 123
0
votes
1
answer
104
views
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
3
votes
1
answer
151
views
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:
$$A:=\left\{p\in\mathcal{P}...
- 291
2
votes
1
answer
237
views
Is the following set compact w.r.t. the Wasserstein distance?
Fix a finite first moment probability measure $q\in\mathcal{P}_1(\mathbb R ^d)$, and real numbers $K,M,R$. Consider the following set:
$$A:=\left\{p\in\mathcal{P}_1(\mathbb R ^d): \int |x|dp\leq K, \...
- 291
1
vote
1
answer
118
views
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
4
votes
1
answer
231
views
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 ...
- 73
2
votes
0
answers
59
views
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 ...
- 21
2
votes
1
answer
203
views
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
1
vote
1
answer
127
views
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
3
votes
1
answer
269
views
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 ...
- 305
3
votes
0
answers
69
views
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 $p_0$ ...
- 131
0
votes
0
answers
55
views
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: $$\left\{ B_p (r) : r>0, p=\sum_{i=1} ^n \alpha_i \delta_{x_i},\text{ ...
- 291
1
vote
1
answer
83
views
Lipschitz approximation of a probability measure with finite $1$-st moment by the ones with finite $p$-th moment
For $p \in [1, \infty)$, let $\mathcal P_p (\mathbb{R^d})$ be the space of Borel probability measures on $\mathbb R^d$ with finite $p$-th moment. We endow $\mathcal P_p (\mathbb{R^d})$ with the ...
- 825
9
votes
2
answers
471
views
Proving the inequality involving Hausdorff distance and Wasserstein infinity distance
Prove the inequality
$$d_{H}(\mathrm{spt}(\mu),\mathrm{spt}(\nu))\leq W_{\infty}(\mu,\nu)$$
where $d_H$ denotes the Hausdorff distance between the supports of the measures $\mu$ and $\nu$, and $W_\...
- 99
3
votes
0
answers
230
views
Are 1-Wasserstein and 2-Wasserstein distances between multivariate normal distributions equivalent?
The $p$-Wasserstein between two measures $\nu_1$ and $\nu_2$ on $X$ is given by
$$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
0
votes
0
answers
114
views
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 ...
- 54
2
votes
0
answers
95
views
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 ...
- 657
0
votes
1
answer
210
views
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
3
votes
0
answers
88
views
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
4
votes
0
answers
148
views
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 $X=\mathbb{R}^d$)...
- 91
3
votes
0
answers
137
views
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
3
votes
0
answers
66
views
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 $\gamma_t:[0,1]\to M$ ...
- 131
1
vote
1
answer
86
views
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, ...
- 825
3
votes
1
answer
162
views
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 ...
- 33
3
votes
0
answers
66
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
$$...
- 51
4
votes
1
answer
301
views
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
2
votes
0
answers
92
views
Construct a 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 \...
- 503
6
votes
2
answers
2k
views
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\...
- 5,380
0
votes
1
answer
54
views
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} \...
- 825
4
votes
2
answers
485
views
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_{\...
- 825
0
votes
1
answer
88
views
Can we lower bound this entropy by $\int_{\mathbb R^d} \rho^k (x) \, \mathrm d x$ and $\int_{\mathbb R^d} |x|^2\rho (x) \, \mathrm d x$?
We define $U : [0, \infty) \to [0, \infty)$ by $U(0) := 1$ and $U (s) := s \log s + (1-s)$ for $s >0$. Then $U$ is strictly convex. The minimum of $U$ is $0$ and is attained at $s=1$. Let $\mathcal ...
- 825
4
votes
2
answers
256
views
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
$$
\...
- 825
3
votes
1
answer
145
views
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
4
votes
0
answers
197
views
Compactness of the unit ball in the space of Radon measures w.r.t. the Kantorovich-Rubinstein norm
This question was posted previously but has not attracted any responses so I am repharising it in a slightly different language hoping to reach a wider community
Let $(X,d)$ be a pointed metric space ...
- 437
2
votes
1
answer
385
views
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 ...
- 657
4
votes
2
answers
1k
views
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 ...
- 407