# 1-Wasserstein distance between two multivariate normal

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

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

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... Jan 24 '18 at 21:51
• Are you interested in explicit formulas for normal distributions only, or for other specific measures as well? Nov 9 '18 at 15:02

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 $p \ge 1$). Jan 19 '20 at 20:47