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The $p$-Wasserstein between two measures $\nu_1$ and $\nu_2$ on $X$ is given by

$$d_p(\nu_{1},\nu_{2})=\left(\underset{\pi\in\Gamma(\nu_{1},\nu_{2})}{\inf}\int_{\mathbf{\mathcal{X}}^{2}}d(x,y)^p\pi(dx,dy)\right)^{\frac 1p}$$

where $\Gamma(\nu_{1},\nu_{2})$ is the set of all couplings between $\nu_1$ and $\nu_2$. For $X=\mathbb{R}^d$ and $d$ being the euclidean distance the optimal transport between $\nu_{1}=N(m,V)$ and $\nu_{2}=N(n,U)$ is well known for $p=2$ see e.g. On Wasserstein geometry of the space of Gaussian measures by Asuka Takatsu. However what is known for $p=1$ for the Euclidean distance or other "reasonable" metrics? I am interested in explicit formulas or sharp bounds.

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    $\begingroup$ Interested in this question as well... $\endgroup$
    – Kashif
    Commented Jan 24, 2018 at 21:51
  • $\begingroup$ Are you interested in explicit formulas for normal distributions only, or for other specific measures as well? $\endgroup$
    – leo monsaingeon
    Commented Nov 9, 2018 at 15:02
  • $\begingroup$ Can't one use the fact that the optimal map between Gaussians is known explicitly? and this map does not depend on the exponent $p>1$ $\endgroup$
    – leo monsaingeon
    Commented Mar 26 at 14:29
  • $\begingroup$ @leomonsaingeon Could you explain or give reference for why optimal maps do not depend on the exponent? $\endgroup$
    – Vladimir Zolotov
    Commented Apr 4 at 17:12

1 Answer 1

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

when $\lambda_i$ and $\gamma_i$ are the $i^{th}$ eigen values of $U$ and $V$ respectively, $v_1,\ldots,v_d$ and $u_1,\ldots,u_d$ are the corresponding orthonormal basis of eigen-vectors.

See Chafai & Malrieu Lemma 2.4.

Although this bound seems close-in nature to the $p=2$ bound, I'm not sure if it is sharp.

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    $\begingroup$ There is a typo in your bound. The "$uv$" term should be $(1-v_i\cdot u_i)$. Also, note that this bound is an equality when the covariance matrices are equal (infact in this case all $W_p(\nu_1,\nu_2) = \|m-n\|_2$ for all $p \ge 1$). $\endgroup$
    – dohmatob
    Commented Jan 19, 2020 at 20:47
  • $\begingroup$ Fixed. Thanks for pointing it out $\endgroup$
    – Meni
    Commented Jan 20, 2020 at 6:33

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