# 1-Wasserstein distance between two multivariate normal

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.

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

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.
Although this bound seems close-in nature to the $$p=2$$ bound, I'm not sure if it is sharp.
• 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 $p \ge 1$). Commented Jan 19, 2020 at 20:47