# Tag Info

### 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 ...
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### 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 ...
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### 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 ...
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### 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....
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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 ...
• 5,163
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 ...
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### 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 ...
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### 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 ...
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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 ...
• 120k
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 ...
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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 ...
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### 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 ...
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Accepted

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### 1-Wasserstein distance between two multivariate normal

For $p=1$ one can bound the 1-Wasserstein metric by $$|m-n| + \sqrt{\sum_{i=1}^{d} \left[ \left( \sqrt{\lambda_i} - \sqrt{\gamma_i}\right)^2 + 2\sqrt{\lambda_i\gamma_i}(1-v_i\cdot u_i) \right]}$$ ...
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### 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 $\Omega$...
• 5,394
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 ...
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### 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 ...
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Accepted

• 120k