Does Wasserstein distance/metric follow triangle inequality in ot.emd2 implementation?

[ogencoglu]

This is a question and not an issue.

I can find sources that show Wasserstein distance indeed follow triangle inequality but does Wasserstein distance/metric implemented by ot.emd2 follow triangle inequality as is?

[rflamary]

Hello emd2 returns the linear OT loss:

• it is equal to $W_1$ when cost matrix M contains euclidean distances (and respect triangular inequality)
• it is equal to $W_2^2$ when cost matrix M contains suqared euclidean distances (and respect triangular inequality whne a square root is added)