Sinkhorn Divergence code does not match referenced paper

[davibarreira]

Describe the bug

I'm opening as a bug, but the code actually does work. The issue is the definition used for Sinkhorn Divergence.
Basically, the Sinkhorn Divergence code does not match the formula in the referenced paper (["Learning Generative Models with Sinkhorn Divergences", 2017])(https://arxiv.org/pdf/1706.00292.pdf).

Code sample and Expected behavior

Here is a piece of the code from empirical_sinkhorn_divergence:
sinkhorn_div = sinkhorn_loss_ab - 0.5 * (sinkhorn_loss_a + sinkhorn_loss_b).

To match the sinkhorn divergence formula from the paper, the code should probably be:
sinkhorn_div = 2* sinkhorn_loss_ab - (sinkhorn_loss_a + sinkhorn_loss_b).

This is a minor issue, but perhaps the documentation should address this difference.

Another issue is that the sinkhorn_loss returns

$$W &= \min_\gamma <\gamma,M>_F + reg\cdot\Omega(\gamma)$$

While in the paper, the sinkhorn cost is only

$$W &= \gamma* <\gamma,M>_F,$$

where \gamma* is the optimal plan for the regularized problem. In other words, the regularization term is only used to find the optimal plan and is then discarded.

[rflamary]

This is a good point. Note that the Sinkhorn divergence in the paper below with exactly the same notation but different definition (and is studied theoretically) has the 1/2 scaling and contains the entropic regularization term.

http://proceedings.mlr.press/v89/genevay19a/genevay19a.pdf

Also note that we actually return only the linear part in the sinkhorn function so we do not implement thet paper above either ;).

I agree that we definitely need a better documentation here. Feel free to propose a PR.

[rflamary]

Note that we are currently working ona new API that will aloow to recover le linear on complete loss and implement properly the different variants of sinkhorn divergence.

[davibarreira]

Thanks for the answer!