For 2-Wasserstein distance between two normal distributions, there is a nice closed-form formula.
Now, I want to find the distance between a normal distribution $\mathcal{N}(\mu, \sigma^2)$ and, say, an exponential distribution $\exp(\lambda)$.
There's a formula to calculate the 2-Wasserstein distance between continuous one dimensional distributions. However, it involves integral over the inverse CDF of a random variable, which cannot be computed for normal distribution.
If I follow the definition of Wasserstein distance, then the problem is about maximizing $E[XY]$, for which I have no clue about how to solve.
I want to ask: is there a way to compute 2-Wasserstein distance between normal distribution and other continuous random variable, say something from exponential family? Is it possible to generalize the computation to high-dimensional distributions?
Any suggestion is appreciated. Thanks!