[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       +/-   ##
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- Coverage   96.49%   83.70%   -12.79%     
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  Files          67       67               
  Lines       14663    14708       +45     
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- 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.