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18 votes

Manifold of probability measures: connections between two types of metrics

In response to the critical comments below I revised my answer. Hope this is more helpful! (1) Two kinds of metrics are defined on generally different spaces. It is not fair to compare these two ...
Henry.L's user avatar
  • 8,011
15 votes

Do distance functionals separate probability measures?

No. Suppose that $\Omega$ consists of four points arranged in a square, where adjacent points have distance 1 between them and opposite points have distance 2. Specifically, if the points are labeled ...
George Lowther's user avatar
13 votes

The geometric median of a triangle

Not an answer, but this paper Carlsson, John Gunnar, Fan Jia, and Ying Li. "An approximation algorithm for the continuous $k$-medians problem in a convex polygon." INFORMS Journal on Computing 26.2 ...
Joseph O'Rourke's user avatar
11 votes

Manifold of probability measures: connections between two types of metrics

Edit (June 2022): Jun Zhang and I wrote a survey paper on some interactions between these two fields which expands on what I mentioned here. You can find the paper at the following link: https://arxiv....
Gabe K's user avatar
  • 5,394
11 votes
Accepted

How does Otto theory work in this example of Wasserstein a.c. curve of probabilities?

This is the typical example of a curve of measures which is as nice as it gets for any "linear structure" (e.g. TV norm or whatever) but is somehow the worst case scenario for quadratic Wasserstein ...
leo monsaingeon's user avatar
10 votes
Accepted

Comparison of Information and Wasserstein Topologies

It is not the case that the Fisher-Rao distance dominates the Wasserstein distance. For instance, it fails for univariate normal distributions $\mathcal{N}(\mu,\sigma)$. In particular, the Wasserstein ...
Gabe K's user avatar
  • 5,394
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 ...
leo monsaingeon's user avatar
8 votes

Manifold of probability measures: connections between two types of metrics

Just a quick follow-up: Very recently (well, actually in 2015) three teams came up independently and almost simultaneously with the same construction of a new "optimal-transport-like" distance on the ...
leo monsaingeon's user avatar
8 votes
Accepted

Local Lipschitzness of parameterization of Gaussians in Wasserstein space

$\newcommand{\R}{\mathbb R}\newcommand{\tr}{\operatorname{tr}}$The answer is yes. Indeed, it is easy to see (cf. e.g. Proposition 7 or the beginning of its proof) that the Wasserstein distance between ...
Iosif Pinelis's user avatar
7 votes
Accepted

Inf of Jensen's inequality

Jensen's inequality $\int c(f)\,d\mu \ge c(\int f\,d\mu)$ is always equality when $f$ is constant, so you know that taking $dz/dt$ constant would saturate it. That means $z(t)$ has to be linear, and ...
Nate Eldredge's user avatar
7 votes
Accepted

About the metrizability of the space of Probability measures $\mathcal{P}(S)$

I'm not really sure what Villani wrote in his monograph, but it is true that one needs to prove that the weak topology is induced by a distance, as a priori it could be another topology with the same ...
Nicola Gigli's user avatar
7 votes

Do distance functionals separate probability measures?

On the positive side, the answer is affirmative if $\Omega$ is the unit interval $[0,1]$ with its standard distance. In this case $\phi_\mu$ is a convex $1$-Lipschitz function (in fact, it is also ...
Pietro Majer's user avatar
7 votes
Accepted

Fokker-Planck: equivalence between linear spectral gap and nonlinear displacement convexity?

One has the ordering of the three $\lambda$'s, i.e. $$ \lambda_{\text{convex}} \leq \lambda_{\text{LSI}} \leq \lambda_{\text{SG}}, $$ where $\lambda_{\text{convex}}$ is the one from convexity, $\...
André Schlichting's user avatar
6 votes
Accepted

Wasserstein interpolation between two probability measures on a metric space

The following discussion is based on the book Gradient Flows by Ambrosio, Gigli, and Savare (2008). Consider $p$-Wasserstein distance with $p>1$ on a Hilbert space (for the sake of uniqueness). ...
O. Richard's user avatar
6 votes
Accepted

Universal decay rate of the Fisher information along the heat flow

The result actually holds with $C=d/2$ on any compact Riemannian manifold with a non-negative Ricci curvature. This can be seen by integrating the Li-Yau inequality: Theorem 1.1 in On the parabolic ...
Fabrice Baudoin's user avatar
6 votes
Accepted

Effect of perturbing the atoms of a measure on the Wasserstein distance

There is a nontrivial counterexample for $N=2$, $p=1$, and $X=\mathbb{R}$. Pick $x_1=-2$, $x_2=2$, $x'_1=-1$, $x'_2=1$ and $p_1=4/5$ and $p_1'=1/5$. Then $2.2=W_1(G,G'')<W_1(G,G')=2.4$. (I hope I ...
S.Surace's user avatar
  • 1,675
6 votes
Accepted

What is the intuition behind the Kantorovich potential in optimal transport?

I recommend the interpretation with bakeries and cafes! In Villani's "Optimal Transport Old and New" in Chapter 5 "Cyclic monotonicity and Kantorovich duality you'll find this: I shall ...
Dirk's user avatar
  • 12.4k
6 votes
Accepted

Wasserstein convergence of "series expansion'' of probability measure

It is true and clear if the metric space $X$ has a finite diameter, but false in general: Take $\beta_i=2^{-i}$ and $\mu_i$ the point mass at $3^i$. Details: In the case $D=$diam$(X)<\infty$, write ...
Yuval Peres's user avatar
6 votes

Weak convergence + convergence of the norm implies strong convergence in Orlicz spaces

There's a famous trick attributed to Minty (sometimes also to Browder and Lions) which allows to recover at least a.e. convergence when you're able to pass to the limit through an increasing non-...
Ayman Moussa's user avatar
  • 2,865
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 ...
leo monsaingeon's user avatar
5 votes

Wasserstein distance and the Kantorovich-Rubinstein duality

You might look at Chapter 3 of my book Lipschitz Algebras (second edition). The Banach space ${\rm Lip_0}(X)$ is already the dual of the space of finitely supported measures on $X$ satisfying $\mu(X) =...
Nik Weaver's user avatar
  • 42.4k
5 votes

1-Wasserstein distance between two multivariate normal

For $p=1$ one can bound the 1-Wasserstein metric by $$|m-n| + \sqrt{\sum_{i=1}^{d} \left[ \left( \sqrt{\lambda_i} - \sqrt{\gamma_i}\right)^2 + 2\sqrt{\lambda_i\gamma_i}(1-v_i\cdot u_i) \right]}$$ ...
Meni's user avatar
  • 181
5 votes

Bounding probability densities on a Wasserstein-2 geodesic

It turns out that for most convex domains $\Omega$, one can find smooth probability measures $\rho_0$ and $\rho_1$ which are supported on $\Omega$, have strictly positive density everywhere on $\Omega$...
Gabe K's user avatar
  • 5,394
5 votes
Accepted

How does a statistical divergence change under a Lipschitz push-forward map?

If $d(\mu,\nu)$ is taken to be the total variation metric, then Lipschitz and metric properties don't matter. This is due to a "data processing inequality" of sorts: applying a transformation can only ...
Aryeh Kontorovich's user avatar
5 votes

Open Questions about Wasserstein Space and PDE

I work in a bit different field too, but your question is quite interesting for me. So I have looked up in the literature just for curiosity. Since there are (surprisingly) no answers yet, let me ...
Skeeve's user avatar
  • 1,277
5 votes
Accepted

How to interpret couplings in optimal transport?

Of the mass $\mu(A)$ in $A$ a fraction $\pi(A \times B)$ is transported to $B$, so you can think of this as a randomized transport map. A basic example to think of is $\mu=\delta_0$ and $\nu=(\delta_1+...
Yuval Peres's user avatar
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)=...
leo monsaingeon's user avatar
5 votes
Accepted

For diffeomorphism $f$, if $X$ and $f(X)$ are both Gaussian, then $f$ is affine

$\newcommand\R{\mathbb R}$Ben McKay's idea stated in the comment above is a natural one for a counterexample. Indeed, for $(x,y)\in\R^2$, let $$f(x,y):=f((x,y)):= \left(x \cos \left(r^2\right)-y \sin \...
Iosif Pinelis's user avatar
5 votes

Mass transportation proof of the Gaussian isoperimetric inequality?

See e.g. Section 2.1 "Talagrand's transport inequalities and Gaussian dimension-free concentration" of Gozlan's survey. Theorem 2.3 there is Talagrand's result that the standard Gaussian ...
Iosif Pinelis's user avatar

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