All Questions
25 questions
8
votes
1
answer
727
views
continuity of the Boltzmann entropy in the Wasserstein metric
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
$$
\...
- 5,380
8
votes
3
answers
936
views
Question about Wasserstein metric
Let $\mu$ and $\nu$ be two probability measures on $\mathbb R^n$ with finite first moment. Denote by $d:=W_1(\mu,\nu)$, where $W_1(\cdot,\cdot)$ stands for the Wasserstein distance of order $1$.
My ...
- 345
4
votes
2
answers
415
views
Effect of perturbing the atoms of a measure on the Wasserstein distance
Let $(X,d)$ be a metric space, $x_1,\ldots,x_N\in X$ and $x_1',\ldots,x_N'\in X$ be atoms, and $G=\sum_{i=1}^Np_i\delta_{x_i}$, $G'=\sum_{i=1}^Np_i'\delta_{x_i}$, and $G''=\sum_{i=1}^Np_i'\delta_{x_i'}...
- 710
4
votes
1
answer
444
views
PDE-Based Triangle Inequality for Optimal Transportation
Suppose $\Omega$ is a suitably regular domain in $\mathbb{R}^n$ and $\rho_0,\rho_1\in\textrm{Prob}(\Omega)$. Benamou and Brenier showed that the $L_2$ transportation distance between $\rho_0$ and $\...
- 705
4
votes
2
answers
256
views
Are the sublevel sets of Boltzmann entropy compact in Wasserstein distance?
For Lebesgue-absolutely continuous probability measures $\rho\ll \mathcal{L}^d$ in the whole space $\mathbb{R}^d$ with finite second moments (i-e $\rho\in \mathcal{P}^2_{ac}(\mathbb{R}^d)$), let
$$
\...
- 825
4
votes
1
answer
360
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}...
- 345
4
votes
0
answers
757
views
Tangent space and gradient on subspace of Wasserstein space given by finitely supported measures
Let $\mathcal{P}_2(M)$ be the 2-Wasserstein space over some Riemannian manifold $(M,g)$ (connected, complete, and without boundary). Let $\mathcal{FP}_2(M,n)$ be the subspace of probability measures ...
- 1,675
3
votes
2
answers
295
views
Wasserstein convergence of "series expansion'' of probability measure
Let $X$ be a Polish space and let $(\mu_i)_{i=1}^{\infty}$ be a sequence of probability measures in the Wasserstein space $\mathcal{P}(X)$ on $X$. Let $(\beta_i)_{i=1}^{\infty}$ be a summable ...
- 5,405
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 ...
- 6,853
2
votes
1
answer
309
views
Density in Wasserstein space
I am wondering whether the following result is true:
Let $\mathcal W_p(\mathbb R^d)$ be the Wasserstein space of order $p$ and let $\eta$ and $\gamma$ be two probability measures in $\mathcal W_p(\...
- 325
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 $\pi$, i.e., if for all $\rho$,
$$H_\pi(\rho):=\int \rho(x)\log\frac{\rho(x)}{\pi(x)}dx \leq \frac{1}{2\beta}\int \rho(x)\left\|\nabla \log\frac{\...
- 21
2
votes
0
answers
49
views
$\sigma$-compactness of probability measures under a refined topology
Denote Polish spaces $(X, \tau_x)$ and $(Y, \tau_y)$, where $X$ and $Y$ are closed subsets of $\mathbb{R}$. Consider a Borel measurable function $f: (X \times Y, \tau_x \times \tau_y) \rightarrow \...
- 195
2
votes
0
answers
302
views
Simplify Kantorovich–Rubinstein duality when distributions share a common marginal
Consider the product of two metric spaces $X\times Y$, and two probability distributions $\mu$ and $\nu$ on this product space. By the Kantorovich-Rubinstein duality, I can write the Wasserstein-1-...
- 53
1
vote
1
answer
1k
views
Closed-form upper-bounds for Wasserstein distance between finite measures
Let $x_1,\dots,x_n,y_1,\dots,y_n\in \mathbb{R}$ and such that $x_i\neq x_j$ and $y_i\neq y_j$ if $i\neq j$. Let $a,b$ be elements of the probability n-simplex. Define the measures $\mu\triangleq \...
- 5,405
1
vote
1
answer
241
views
Continuity of pushforward operation
Let $X$ and $Y$ be compact metric spaces and let $f,g:X\rightarrow Y$ be $\epsilon$-uniformly close; i.e.:
$$
\sup_{x \in X} d_Y(f(x),g(x))<\epsilon.
$$
Then, are their push-forwards close in ...
- 5,405
1
vote
0
answers
169
views
Optimal transport-like problem where the objective depends on conditional probability distribution
$\DeclareMathOperator\marg{marg}$I would like to know if the following problem can be studied as an optimal transport problem, possibly imposing additional assumptions on the data.
Consider two sets $\...
1
vote
1
answer
719
views
Transport of measure
Let's disintegrate $\mu$ and $\nu$, two probabilities on $\mathbb{R}^{d}$ , according to
$$
\pi_{k} (x_{1},...,x_{d}) = (x_{k},...,x_{d})
$$
We get a family of measures and each measure $\mu_{k,d}^{+...
- 179
0
votes
2
answers
201
views
Couplings as generalized functions
I've been casually reading about optimal transport, and I was intrigued by the Wasserstein metric, in which we define the distance between two measures $\mu$ and $\nu$ on a metric space $X$ by
$$
W_p(\...
- 123
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
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
0
votes
1
answer
86
views
Is integration against an indicator Wasserstein-Continuous
Let $\mathcal{P}_p(X)$ denote the Wasserstein space over a compact metric space $X$, and $1\leq p<\infty$. Fix a non-empty closed subset $C\subseteq X$. Then is the map:
$$
\mathbb{P} \mapsto \...
- 5,405
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
0
votes
0
answers
87
views
How does one define weak convergence of probability measures in $L^{\infty}(\Omega)$?
I am reading the following article and on page 9/17 (above Eqn (4.9)) the authors state that if $\gamma_{\epsilon_k}|\_G_{\delta}\times \Omega\to \gamma|\_G_{\delta}\times \Omega$ as $\epsilon_k\to 0$ ...
- 547
0
votes
0
answers
75
views
Optimizer of a semi-discrete optimal transport problem
Provided two probability distributions $\mu(dx)=\rho(x)dx$ and $\nu(dx)=\sum_{i=1}^n p_i\delta_{y_i}(dx)$ that are supported on some measurable set $\Omega\subset\mathbb R^d$, we consider the semi-...
user128095
0
votes
0
answers
184
views
Reference request: density of $C_c^{\infty}(\mathbb R^d)$ in $L^2(\mathbb R^d,d\rho)$
My question is motivated by an optimal transportation approach to PDE's and gradient flows in metric spaces (see e.g Otto's geometry of dissipative evolution equations: the porous media equation and ...
- 5,380