from the source to the target that passes through a relay measure in the middle (see Figure \[fig:ship1\]). We solve it by VWBs. Our discussion comes directly from the conclusions in \[sec:metric\_property\]:

\[the:vwb\_2metric\] As a special case of Corollary \[the:vwb\_nmetric\], $\VWBL{\NU}(\MUNL{1:2})$ induces a (2-)metric between $\MUL{1}$ and $\MUL{2}$. It is lower-bounded by $\frac{1}{4}\WLU{2}{2}(\MUL{1},\MUL{2})$ when

$|\MUL{i}| = |\NU| = K$.

Appendix reveals the proof.

Then, we can use a VWB to connect two measures and regard the total WD as an approximation to the true WD between them. We name it the *variational Wasserstein distance*, or *VWD*: $$\label{eq:vwb_ship}
\begin{split}
    \VWLU{2}{2}(\MUL{1}, \MUL{2}) &\eqdef 4 \VWBL{\NU}(\MUNL{1:2})\\ 
    & =\underset{\NU \in \MUS(\Y)}{\inf} 2\ \WUL{2}{2}(\MUL{1}, \NU) + 2\ \WUL{2}{2}(\MUL{2}, \NU). \nonumber
\end{split}$$ We use the toy data above to evaluate the approximation against the number of supports, $K$. The two Gaussian measures share the same covariance matrix; their means differ by 1. Thus, the true WD is $1$. We use the results from linear programming (LP) and Sinkhorn algorithms for reference. Figure \[fig:ship2\] shows that VWD is still accurate with few supports. For each number of supports in the experiments, we run our algorithm 10 times with different random initial locations. We draw the error band 