10
votes
Accepted
Upper bound total variation by Wasserstein distance for continuous distance
No. One should realize that the transportation and the total variation distances metrize two quite different topologies. Even if the measures are equivalent (i.e., absolutely continuous with respect ...
- 17k
9
votes
Proving the inequality involving Hausdorff distance and Wasserstein infinity distance
EDIT: answer 2 below is completely false, as pointed out by the OP. However this is such a typical example of wishful thinking that I believe it is worth leaving for the posterity. (I'll record it ...
- 5,380
7
votes
Accepted
Is the following set compact w.r.t. the Wasserstein distance?
Unfortunately not. Take $q = \delta_0$ and $p_n = (1-n^{-1})\delta_0 + n^{-1}\delta_n$. Then $p_n \in A$ (with say $K = M = R = 1$) and $p_n \to \delta_0$ weakly but not in $\mathcal{P}_1$...
- 10.3k
6
votes
Accepted
Perturbation of Wasserstein distance: looking for references
You can find this in Villani's "small book", Theorem 8.13 in [Villani, C. (2003). Topics in optimal transportation (Vol. 58). American Mathematical Soc.]
I can also recommend looking at ...
- 5,380
5
votes
1-Wasserstein distance between two multivariate normal
For $p=1$ one can bound the 1-Wasserstein metric by
$$|m-n| + \sqrt{\sum_{i=1}^{d} \left[
\left(
\sqrt{\lambda_i} - \sqrt{\gamma_i}\right)^2 + 2\sqrt{\lambda_i\gamma_i}(1-v_i\cdot u_i)
\right]}$$
...
- 203
5
votes
Upper bound total variation by Wasserstein distance for continuous distance
If both probability measures have smooth densities, the total variation can be bounded by Wasserstein (even by a weaker metric Levy-Prokhorov), see the 2017 paper A novel approach to Bayesian ...
- 51
5
votes
Accepted
Gradient of Wasserstein distance in the sense of Otto's calculus
Yes this is true, formally this follows by the envelope theorem. In an abstract and very smooth setting, the envelope theorem says that for an objective functional depending on a parameter $t$
$$
F(t)=...
- 5,380
5
votes
Accepted
Wasserstein distance and put function
$\newcommand\R{\Bbb R}\newcommand\LS{\mathsf{LS}}$For real $x$ and $y$ such that $x<y$,
$$\begin{aligned}
P_\mu(y)-P_\mu(x)&=\int_\R[(y-t)_+-(x-t)_+]\mu(dt) \\
&=\int_\R[(y-x)\,1(t\le x)+(...
- 128k
4
votes
Accepted
Wasserstein distance between product measures
$\newcommand{\de}{\delta}\renewcommand{\S}{\mathcal S}\newcommand{\T}{\mathcal T}$The answer to your question is negative if $p<2$.
Indeed, let $\nu_i=\de_0$ for all $i$, where $\de_a$ is the Dirac ...
- 128k
4
votes
Accepted
2-Wasserstein metric on convolution of probability distributions
The answer is yes to the second question, and hence yes to the first question as well.
Indeed, it is easy to check that the functions $f$ and $g$ given by
$$f(t):=\max(0,1-|t|)$$
and
$$g(t):=\sum_{k=-\...
- 128k
4
votes
Accepted
Does complete and separable Wasserstein space imply a complete base space?
$\newcommand\de\delta\newcommand\ep\varepsilon$The conjecture is true.
Indeed, let $(z_n)$ be a Cauchy-convergent sequence in $(Z,d)$. Then
$$d_{W_p}(\de_{z_m},\de_{z_n})=d(z_m,z_n)\to0$$
(as $m,n\to\...
- 128k
4
votes
Accepted
Compactness with respect to topology induced by total-variation distance
$\newcommand\vpi\varphi\newcommand\ep\varepsilon\newcommand\R{\Bbb R}$What you "need to show" is of course false in general. E.g., suppose that $d=1$ and $\varphi=1_{[0,1]}$. Then your set ...
- 128k
3
votes
Accepted
Is there $\varepsilon \in (0, 1)$ such that $\sup_{t \in [0, \varepsilon]} [\ell_t]_\beta < \infty$?
Pick a sequence of functions $g_n$ where (i) $\|g_n\|_\beta=n$ and (ii) $\|g_n-\mathbf 1\|_\infty\le 3^{-n}$. For example
$$
g_n(x)=\begin{cases}
1+n(a_n-x)^\beta&\text{if $0\le x\le a_n$};\\
1&...
- 23.2k
3
votes
Wasserstein distance between product measures
From the previous answer, it follows that the inequality is true when $p\ge 2$. Let $X_i, Y_i$ be the optimal choice of random variables for which
$$\|X_i-Y_i\|_p=W_p(\mu_i,\nu_i) \ \forall i=1,\ldots,...
- 407
3
votes
An inequality involving the Wasserstein distance and chi-squared distance
$\newcommand{\N}{\mathbb N}$By the known expression for the Wasserstein distance,
\begin{equation*}
\begin{aligned}
W_1(p,q)&=\int_0^\infty dx\,\Big|\sum_{n>x}(p_n-q_n)\Big| \\
&\...
- 128k
3
votes
Accepted
The uniqueness of Barycenters in the Wasserstein space
Uniqueness follows directly from proposition 3.3 on the previous page. For completeness, a shortened version containing the relevant parts:
Suppose $\mu$ and $\nu$ are probability measures on $\...
3
votes
Accepted
Is this set $\sigma$-compact in the Wasserstein space?
It seems $A$ is not $\sigma$-compact. Check my argument as I have never worked with Wasserstein distances. Let $d=1$, let $q$ be the measure supported in $\{0\}$, so that $W_1(q,\mu)=\int|x|d\mu$ for ...
- 10.6k
2
votes
Upper bound total variation by Wasserstein distance for continuous distance
To complement the existing answers, the following example shows that a bound
$$
\|\mu - \nu\|_{1} ≤ C W_2(\mu, \nu) \tag{1}
$$
does not hold even when we restrict the densities to smooth, globally ...
- 211
2
votes
Accepted
Bounding $2$-Wasserstein distance and the $L^1$ distance
Referring to the proof of Prop 3.21 in Malrieu 2001, after the triangle inequality is applied twice, two of the terms are bounded via the upper bound $$
W_2(u_t,u_t^{(1,N)}) \vee W_2(\mu_{1,N},\bar{u})...
- 6,045
2
votes
Accepted
Getting Wasserstein closeness from a derivative estimate
Assuming the support of both measures is fixed, it is enough to bound $W_1(\mu,\nu)$. That is, we want to bound $\vert \int f d\mu -d\nu \vert$ for any $1$-Lipschitz function $f$.
Take a triangulation ...
- 2,772
2
votes
Accepted
Are the sublevel sets of Boltzmann entropy compact in Wasserstein distance?
$\newcommand{\R}{\mathbb R}\newcommand{\H}{\mathcal H}$Building up on previous comments, and slighlty elaborating. The answer to your question, as is, is NO. Recall that convergence in the Wasserstein ...
- 5,380
2
votes
Accepted
Some continuity issues of the optimal transport map (Brenier map)
$\newcommand{\si}{\sigma}\newcommand{\W}{\mathcal W}\newcommand{\R}{\mathbb R} $Yes, the map
\begin{equation*}
\mu \mapsto J(\mu):=\int_\R T^\si_\mu(x+y)\exp(-y^2/2\si^2)\,dy \tag{10}\label{10}
\...
- 128k
2
votes
Accepted
Connection between Wassertein-2 metric and difference in variance
Let $\mu_k$ and $\nu_k$ denote the $k$th moments of $\mu$ and $\nu$ respectively.
Write
$$W_p(\mu,\nu)=\inf\{\|X-Y\|_p\colon X\sim\mu,Y\sim\nu\},$$
where $\|Z\|_p:=(E|Z|^p)^{1/p}$.
$\newcommand\si\...
- 128k
2
votes
Accepted
Unique coupling
The only way this can happen is the situation that you described, where at least one of the measures gives full measure to a single point. In fact, the proof below does not require the two Polish ...
- 23.2k
2
votes
Unique coupling
For two measurable sets $A,B$, let $p=\mu(A)$ and $q=\nu(B)$. Consider any coupling of a Bernoulli$(p)$ and a Bernoulli$(q)$, say, $C:\{0,1\}^2 \to [0,1]$. Then we can find a coupling $(X,Y)$ of $\mu$...
- 1,724
1
vote
Accepted
Lipschitz approximation of a probability measure with finite $1$-st moment by the ones with finite $p$-th moment
$\newcommand{\R}{\mathbb R}$This is only a partial answer in the sense that it will provide $\frac 1p$-Hölder maps, not Lipschitz. I still hope it can help.
Fix a big radius $R>0$ (I like $R\to\...
- 5,380
1
vote
Accepted
Gradient flows: evolution of geodesics
As currently asked the answer is NO, because your desired upper bound already fails for $t=0$ (or equivalently, $t=1$). Indeed, it is well understood that the small-time deviation along the heat flow, ...
- 5,380
1
vote
Are the sublevel sets of Boltzmann entropy compact in Wasserstein distance?
It is known that the sublevel sets of the relative entropy are tight when the reference measure is finite, and in fact are also compact in the topology of setwise convergence (which is stronger than ...
- 660
1
vote
Lipschitz-type inequalities for Markov kernels
Assuming that your distance is convex, the question reduces just to the case when both $\mu$ and $\nu$ are delta measures, and amounts then to the inequality
$$\tag {$\star$}
d(\delta_x,\delta_y) \le ...
- 17k
1
vote
Accepted
Is the Wasserstein distance to the empirical measure minimized by the underlying distribution?
No. E.g., let $N=1$ and suppose that $X:=X_1$ has a nondegenerate zero-mean distribution $\mu$ such that $E|X|^p<\infty$. Let $Y$ be an independent copy of $X$.
Then the expected $\mathcal W_p$-...
- 128k
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