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Let $\mathcal{X},\mathcal{Y}\subset \mathbb{R}^d$ and $\mu,\nu$ Borel probability measures on $\mathcal{X},\mathcal{Y}$ respectively. The Wasserstein-2 distance is defined as

$$ W_2(\mu,\nu):=\inf_{\pi\in\Pi(\mu,\nu)}\Big(\int_{\mathcal{X} \times \mathcal{Y}}|x-y|^2d\pi(x,y)\Big)^{1/2}. $$ Where $\Pi(\mu,\nu)$ is the space of all joint distributions with marginals $\mu,\nu$. Recall $W_2$ ( on the space of borel probability measures with finite second moments ) satisfies the properties of a metric. Furthermore a sequence of probability measures converges in this metric if and only if they converge weakly and their second moments converge.

$\underline{Question :}$ I have heard people say that the Wasserstein `lifts' the underlying metric. Are they referring to anything specific, or is this just a fancy way to say what I wrote above?

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    $\begingroup$ Yes and no. A lift is a "generic" operation where you have a "lower" structure $B$ and an "upper" structure $T$ (like an algebra over the lower structure), where you find/have an operator $L$ which maps functions $f\colon B \rightarrow X$ to functions $L(f) \colon T \rightarrow X$. So the "lift" operator "imports" that $f$ function into the $T$ space. $\endgroup$
    – nomen
    Commented Jul 27, 2020 at 18:14
  • $\begingroup$ 2/In your case, apparently the operator $L(f) = \inf \left( \int_{X \times Y} f^2 d\pi(x,y) \right)^{1/2}$ satisfies the "functorial properties" it needs to have to be a "lift". (I do believe it, since it looks like a plausible isometry, but I haven't gone through the details to verify it...) $\endgroup$
    – nomen
    Commented Jul 27, 2020 at 18:16

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To describe lifting you have to say how the original underlying metric sits inside the new problem.

That is you consider the case when $\mu$ and $\nu$ are localized around points $p$ and $q$. What you can do now is compute the distance between $p$ and $q$ for the underlying metric. You can also compute the distance between $\mu$ and $\nu$. If those two line up, then that is what is meant by lifting.

The picture is you have the world of probability above (because it is strictly richer) and the world of points below. But more crucially there is an embedding from below to above, so you can compute solely in the below world or you can lift it and then compute above. You want the results to line up.

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    $\begingroup$ And just to add here the second paragraph in formulas, you have $W_{2}(\delta_{p},\delta_{q})=\|p-q\|$. $\endgroup$
    – Tobsn
    Commented Oct 25, 2020 at 17:52

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