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 ...
- 8,071
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 ...
- 17.1k
12
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....
- 6,001
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 ...
- 5,380
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 ...
- 6,001
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
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 ...
- 5,380
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 ...
- 128k
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 ...
- 29.8k
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 ...
- 660
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 ...
- 60.5k
7
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 ...
- 12.7k
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, $\...
- 1,081
7
votes
Accepted
Does this maximisation problem admit a finite upper bound?
The answer is "yes" and it is quite a nice linear algebra problem, but let me restate it first in a less intimidating way. We'll deal with $\mathbb R^n$ for any finite $n$.
The first thing I ...
- 61.9k
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
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). ...
- 422
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 ...
- 2,703
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 ...
- 1,675
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 ...
- 14.2k
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-...
- 3,425
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
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 $\Omega$...
- 6,001
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) =...
- 42.8k
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
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 ...
- 6,541
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 ...
- 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+...
- 14.2k
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 $t$
$$
F(t)=...
- 5,380
5
votes
Accepted
For diffeomorphism $f$, if $X$ and $f(X)$ are both Gaussian, then $f$ is affine
$\newcommand\R{\mathbb R}$Ben McKay's idea stated in the comment above is a natural one for a counterexample.
Indeed, for $(x,y)\in\R^2$, let
$$f(x,y):=f((x,y)):=
\left(x \cos \left(r^2\right)-y \sin \...
- 128k
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