[WIP] Gaussian initialization for sinkhorn

[rflamary]

Types of changes

This is a first shot for intializing empirical sinkhorn but the computational gain is not very clear on tests I did locally.

The following code

n = 2000 rng = np.random.RandomState(0) x = rng.randn(n, 2) x2 = 10*rng.randn(n//2, 2) x2[:,0]+=2 ot.tic() G, log = ot.empirical_sinkhorn(x,x2, 1, method='sinkhorn_log', warmstart=None, verbose=False, isLazy=False, stopThr=1e-5, log = True) ot.toc() print("Err=",log['err'][-1], "niter=", log['niter']) ot.tic() G2, log2 = ot.empirical_sinkhorn(x,x2, 1, method='sinkhorn_log', warmstart='gaussian', verbose=False, isLazy=False, stopThr=1e-5, log = True) ot.toc() print("Err=",log2['err'][-1], "niter=", log2['niter'])

give sthe following output

Elapsed time : 3.0527355670928955 s Err= 9.441113655553818e-06 niter= 140 Elapsed time : 2.391462564468384 s Err= 9.89690596643644e-06 niter= 110` 

Quite far from the computational gains in the paper. Will investigate it more.

Motivation and context / Related issue

How has this been tested (if it applies)

PR checklist

• I have read the CONTRIBUTING document.
• The documentation is up-to-date with the changes I made (check build artifacts).
• All tests passed, and additional code has been covered with new tests.
• I have added the PR and Issue fix to the RELEASES.md file.

[codecov]

Codecov Report

Merging #555 (b3be5a6) into master (53dde7a) will decrease coverage by 12.79%.
The diff coverage is 91.83%.

Additional details and impacted files

@@ Coverage Diff @@ ## master #555 +/- ## =========================================== - Coverage 96.49% 83.70% -12.79%  =========================================== Files 67 67 Lines 14663 14708 +45 =========================================== - Hits 14149 12312 -1837  - Misses 514 2396 +1882 

[cedricvincentcuaz]

I believe that your implementation is correct, i am not sure how authors handle the bias in empirical_bures_wasserstein_mapping set by default to True in POT.
From the experiments in the paper I would say that gains seem specific to low regimes for the entropic regularization, did you check that ?

[rflamary]

will do. iI have students looking into this. Will come ack to the PR after.