[Math] 1-Wasserstein distance between two multivariate normal

Question

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.

Answer 1

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.