Wasserstein Metric

Today's diary is going to be on my research question because it's been occupying me today. So the immediate goal is to find a way to define "distance" between two gaussian distributions (multivariate). For gaussian distributions, mean vector and the covariance metric can define it. I learned about the Wasserstein metric, also known as "Earth Mover" distance. My next goal is to calculate the Wasserstein metric for multivariate gaussian distributions with different mean and covariance matrix. Wikipedia actually had a formula, but I don't quite buy it yet because I couldn't verify metric symmetry (W(x,y) =W(y,x)).