All Questions
Tagged with optimal-transportation measure-theory
54 questions
2
votes
1
answer
141
views
(Sub)Optimality of random transport
Problem Setup: Consider the intervals IR=[aR,bR] and IB=[aB,bB]. Let FR and FB denote the CDF of distributions with support on the intervals IR and IB. I draw k red and ...
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 μ,ν on Rn. Now consider a map (a priori only measurable, but feel free to add ...
0
votes
0
answers
52
views
Path-homotopy in Wasserstein space
Consider two vector fields b0,b1∈C2([0,1]×Rd;Rd) and the solutions ρ0,ρ1∈AC([0,1];P2(Rd)) to the associated Fokker-Planck equations
$$...
0
votes
0
answers
39
views
Comonotone solution for Optimal Transport problems with supermodular surplus
In Alfred Galichon's book Optimal Transport Methods in Economics the foollowing result is stated for OT problems on the real line.
Theorem 4.3.(i) Assume that Φ is supermodular. Then the primal ...
0
votes
1
answer
104
views
Sequential compactness of a sequence of curves of Borel probability measures
$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\DeclareMathOperator*{\supp}{supp}
\newcommand{\bR}{\mathbb{R}}
\newcommand{\bT}{\mathbb{T}}
\newcommand{\bN}{\mathbb{N}}
\newcommand{\bP}{\mathbb{...
1
vote
1
answer
118
views
Is there ε∈(0,1) such that supt∈[0,ε][ℓt]β<∞?
$
\newcommand{\bR}{\mathbb{R}}
\newcommand{\bT}{\mathbb{T}}
\newcommand{\bN}{\mathbb{N}}
\newcommand{\bP}{\mathbb{P}}
\newcommand{\bE}{\mathbb{E}}
\newcommand{\bF}{\mathbb{F}}
\newcommand{\bD}{\mathbb{...
0
votes
0
answers
55
views
Any useful bases for the topology induced by the t-Wasserstein distance?
I am working on Rd equipped with the usual Euclidean metric. I know of one nice base for Wt, namely: $$\left\{ B_p (r) : r>0, p=\sum_{i=1} ^n \alpha_i \delta_{x_i},\text{ ...
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∈[1,∞), let Pp(Rd) be the space of Borel probability measures on Rd with finite p-th moment. We endow Pp(Rd) with the ...
0
votes
0
answers
114
views
Some stability and estimate of the optimal transport map (Brenier map)
Let μ and ν be two probability measures with finite moments (in P2(R)) equipped with 2-Wasserstein distance. Let Fμ, Fν be their cumulative distribution ...
2
votes
0
answers
95
views
Can we control the Wasserstein metric between μ and ν by their moment difference?
Fix p∈[1,∞). Let (Pp(Rd),Wp) be the Wasserstein space of all Borel probability measures on Rd with finite p-th moment. Let Dp be the collection of ...
0
votes
1
answer
211
views
Some continuity issues of the optimal transport map (Brenier map)
Let μ and ν be two probability measures with finite moments (in P2(R)) equipped with 2-Wasserstein distance. Let Fμ, Fν be their cumulative distribution ...
0
votes
1
answer
88
views
Can we lower bound this entropy by ∫Rdρk(x)dx and ∫Rd|x|2ρ(x)dx?
We define U:[0,∞)→[0,∞) by U(0):=1 and U(s):=slogs+(1−s) for s>0. Then U is strictly convex. The minimum of U is 0 and is attained at s=1. Let $\mathcal ...
4
votes
2
answers
256
views
Are the sublevel sets of Boltzmann entropy compact in Wasserstein distance?
For Lebesgue-absolutely continuous probability measures ρ≪Ld in the whole space Rd with finite second moments (i-e ρ∈P2ac(Rd)), let
$$
\...
4
votes
0
answers
197
views
Compactness of the unit ball in the space of Radon measures w.r.t. the Kantorovich-Rubinstein norm
This question was posted previously but has not attracted any responses so I am repharising it in a slightly different language hoping to reach a wider community
Let (X,d) be a pointed metric space ...
1
vote
0
answers
169
views
Optimal transport-like problem where the objective depends on conditional probability distribution
I would like to know if the following problem can be studied as an optimal transport problem, possibly imposing additional assumptions on the data.
Consider two sets $\...
2
votes
0
answers
49
views
σ-compactness of probability measures under a refined topology
Denote Polish spaces (X,τx) and (Y,τy), where X and Y are closed subsets of R. Consider a Borel measurable function $f: (X \times Y, \tau_x \times \tau_y) \rightarrow \...
4
votes
1
answer
668
views
Optimal Transport: how is this transport map Borel measurable?
I'm reading Theorem 1.17. and its proof at page 14 of Santambrogio's Optimal transport for applied mathematicians. The content is not hard but a little bit long (because of related detail). Please ...
1
vote
1
answer
243
views
Continuity equation for a density of a measure
From the paper of Ambrosio-Crippa, it is known that if β:Rd×[0,T[⟶Rd is suitably regular, then the system
$$
\begin{cases}
\dfrac{\partial\mu}{\partial t}(x,...
3
votes
0
answers
278
views
Radon-Nikodym derivative of vector-valued measure with respect to another vector-valued measure
Let (X,|⋅|) be a Banach space.
I am interested in whether one can extend the definition of the Kullback-Leibler divergence
$$
\text{KL}(\mu \ \Vert \ \nu)
:= \int_{\Omega} \ln\left(\frac{\...
4
votes
1
answer
360
views
Measurability of Markov kernel wrt the Borel σ-algebra generated by the weak topology
Consider two Polish metric probability spaces (A,ΣA) and (B,ΣB), endowed with their Borel σ-algebras. Denote as $\mathcal{P}_\mathcal{B}...
2
votes
0
answers
82
views
Estimate of Wasserstein distance and flow of vector fields under particular assumptions
Let μ be a compactly supported absolutely continuous probability measure. Let v,u be Lipschitz vector fields. For a vector field w recall that Φwt denotes its flow.
A classical estimate ...
0
votes
2
answers
201
views
Couplings as generalized functions
I've been casually reading about optimal transport, and I was intrigued by the Wasserstein metric, in which we define the distance between two measures μ and ν on a metric space X by
$$
W_p(\...
3
votes
2
answers
295
views
Wasserstein convergence of "series expansion'' of probability measure
Let X be a Polish space and let (μi)∞i=1 be a sequence of probability measures in the Wasserstein space P(X) on X. Let (βi)∞i=1 be a summable ...
0
votes
1
answer
275
views
Upper bound on the p-Wasserstein distance Wp(ξ,aξ) for some constant a≠0
Let ξ be a random vector taking values in Rd. Is there an upper bound on the p-Wasserstein distance Wp(ξ,aξ) for some constant a≠0?
I have seen that if $p=...
1
vote
0
answers
130
views
Orthogonality in Wasserstein tangent space for discrete measures with equal mass
Let say I have N discrete probability measures (μ1,...,μN) where each of them has n points in R2 of equal mass.
Let P(X) be the space of these probability measures ...
2
votes
1
answer
173
views
Is there an analogue of transportation-cost inequality under a weighted Log-Sobolev Inequality?
It is known that under the Log-Sobolev Inequality for π, i.e., if for all ρ,
$$H_\pi(\rho):=\int \rho(x)\log\frac{\rho(x)}{\pi(x)}dx \leq \frac{1}{2\beta}\int \rho(x)\left\|\nabla \log\frac{\...
4
votes
1
answer
396
views
Is the optimal transport of radially symmetric measures also radially symmetric?
Let μ and ν be radially symmetric probability measures on Rd. Consider the Kantorovich optimal transport problem between μ and ν, with convex, nonnegative cost. Suppose ...
0
votes
1
answer
86
views
Is integration against an indicator Wasserstein-Continuous
Let Pp(X) denote the Wasserstein space over a compact metric space X, and 1≤p<∞. Fix a non-empty closed subset C⊆X. Then is the map:
$$
\mathbb{P} \mapsto \...
2
votes
0
answers
302
views
Simplify Kantorovich–Rubinstein duality when distributions share a common marginal
Consider the product of two metric spaces X×Y, and two probability distributions μ and ν on this product space. By the Kantorovich-Rubinstein duality, I can write the Wasserstein-1-...
8
votes
0
answers
1k
views
Wasserstein distance and Monge-Kantorovich-Rubinstein duality
The definition of Wasserstein p-distance between two measures μ and ν on a Polish space X is given by
$$
W_p(\mu, \nu)^p = \inf_{\gamma \in \Pi(\mu, \nu)} \int_{X \times X} d(x, y)^p\; d\...
1
vote
1
answer
241
views
Continuity of pushforward operation
Let X and Y be compact metric spaces and let f,g:X→Y be ϵ-uniformly close; i.e.:
supx∈XdY(f(x),g(x))<ϵ.
Then, are their push-forwards close in ...
0
votes
0
answers
87
views
How does one define weak convergence of probability measures in L∞(Ω)?
I am reading the following article and on page 9/17 (above Eqn (4.9)) the authors state that if γϵk|_Gδ×Ω→γ|_Gδ×Ω as ϵk→0 ...
1
vote
1
answer
1k
views
Closed-form upper-bounds for Wasserstein distance between finite measures
Let x1,…,xn,y1,…,yn∈R and such that xi≠xj and yi≠yj if i≠j. Let a,b be elements of the probability n-simplex. Define the measures $\mu\triangleq \...
1
vote
1
answer
247
views
Scaling behavior of Wasserstein distances
Let p>1 and μ≠ν be two probability measures on Ω⊂Rd a bounded set. For α≥0, we let $$C_\alpha(\mu,\nu) = \inf_\sigma \frac{W_p(\mu+\sigma,\nu+\sigma)}...
2
votes
0
answers
146
views
Prove the equicontinuity of a maximizing sequence
Let X be a compact subset of R and c(x1,x2,x3,x4) be a fixed bounded continuous functions on X4. Assume μ,ν are probability measures on X2, and μ⊗ν is the ...
1
vote
1
answer
719
views
Transport of measure
Let's disintegrate μ and ν, two probabilities on Rd , according to
πk(x1,...,xd)=(xk,...,xd)
We get a family of measures and each measure $\mu_{k,d}^{+...
0
votes
0
answers
75
views
Optimizer of a semi-discrete optimal transport problem
Provided two probability distributions μ(dx)=ρ(x)dx and ν(dx)=∑ni=1piδyi(dx) that are supported on some measurable set Ω⊂Rd, we consider the semi-...
6
votes
1
answer
580
views
Why is it difficult to solve the Monge problem directly?
I'm trying to understand something about the Monge problem. The Monge problem is:
Let c(x,y):Rd×Rd→Rd and $$\mathcal{T}(\mu_1,\mu_2) = \{ T: \...
1
vote
0
answers
60
views
Euler-Lagrange Equation for Kantorovich Dual Problem
Given two probability measures μ and ν, the Kantorovich Dual problem for quadratic cost is to
minimize∫ϕ(x)dμ+∫ψ(y)dν
over pairs $(\phi,\psi)\in L^1(d\...
1
vote
0
answers
97
views
Dependency of the Wasserstein metric on its parameters
Let the population on some region Ω⊂Rd be modeled by a density function ρ:Ω→(0,+∞). Provided n≥1 food trucks labeled by their capacity $p_1,\ldots, p_n\in (...
4
votes
0
answers
187
views
Transport Distance between Level Sets of a Convex Function
Suppose I have a well-behaved, strictly convex function f:Rd→[0,∞), and assume that f has its unique minimiser at x=0, with f(0)=0.
For y>0, I define the ...
2
votes
1
answer
317
views
optimal transport, measurable selection
Let X=Y=Rd and let ν be a probability measure on Rd. Consider the collection of probability measure π on X×Y such that π has y-marginal ν:
$$
\Pi(...
4
votes
2
answers
415
views
Effect of perturbing the atoms of a measure on the Wasserstein distance
Let (X,d) be a metric space, x1,…,xN∈X and x′1,…,x′N∈X be atoms, and G=∑Ni=1piδxi, G′=∑Ni=1p′iδxi, and $G''=\sum_{i=1}^Np_i'\delta_{x_i'}...
2
votes
0
answers
116
views
A variant of the optimal transport
Let μ, ν and γ be three probability measures on R. Consider the optimisation problem as follows:
inf(X,Y,Z) E[|Y−Z|2],
where the inf is taken ...
3
votes
0
answers
106
views
Dependency of the Wasserstein distance on the parameter: a differential perspective
Let μ(dx)=∑ni=1piδxi(dx) and ν(dy)=ρ(y)dy be two probability measures on Rd. Consider the 2−Wasserstein distance below:
$$W_2(\mu,\nu)^2 \quad := \quad \inf_{\...
3
votes
0
answers
243
views
Parametric distances on product spaces of measures
Disclaimer: Please excuse my loose language. I'm neither an expert in geometry nor probability. Please ask for clarification if something appears unclear or awkward to you.
Let X be a topological ...
4
votes
0
answers
757
views
Tangent space and gradient on subspace of Wasserstein space given by finitely supported measures
Let P2(M) be the 2-Wasserstein space over some Riemannian manifold (M,g) (connected, complete, and without boundary). Let FP2(M,n) be the subspace of probability measures ...
2
votes
1
answer
309
views
Density in Wasserstein space
I am wondering whether the following result is true:
Let Wp(Rd) be the Wasserstein space of order p and let η and γ be two probability measures in $\mathcal W_p(\...
8
votes
3
answers
937
views
Question about Wasserstein metric
Let μ and ν be two probability measures on Rn with finite first moment. Denote by d:=W1(μ,ν), where W1(⋅,⋅) stands for the Wasserstein distance of order 1.
My ...
8
votes
1
answer
623
views
Completion of spaces of measures w.r.t. weak norms
For the sake of concreteness denote by M0(X) the linear space of all signed Borel measures σ with σ(X)=0 on some metric space (X,d) and fix some base point x0∈X. On this space ...