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Wasserstein distance between two gaussians has a well known closed form solution. Does the same hold for the distance between a Gaussian with a fixed variance(say 1) and the empirical data distribution?

Empirical data distibution defined as: $$ p(x) = \frac{\sum_i \delta (x - x_i)}{n} $$

And the 1-d Gaussian with $\sigma^2 = 1$ and some unknown mean $\mu$ $$ q(x) = \frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{(x-\mu)^2}{2\sigma^2}} $$

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