[WIP] OT barycenters for generic transport costs

[eloitanguy]

Types of changes

• contribute a generalisation of free_support that accepts any cost function (implements this paper)
• contribute a specific version of this function in ot.gmm for fast computation of GMM barycenters
• example for new barycenter function
• example of GMM barycenters
• add summary of features and example in README.md
• allow the barycentric projection solver to accept ground_bary=None and to accept different notions of barycentric projections w.r.t. costs different to L2.
• simple BCD solver with GD on the positions X

Motivation and context / Related issue

How has this been tested (if it applies)

• full coverage in test/test_ot.py and test/test_gmm.py

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

Attention: Patch coverage is 98.31461% with 3 lines in your changes missing coverage. Please review.

Project coverage is 97.14%. Comparing base (cbdf979) to head (371e3e7).

Additional details and impacted files

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##           master     #715    +/-   ##
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  Coverage   97.13%   97.14%            
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  Files         100      100            
  Lines       20369    20545   +176     
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+ Hits        19786    19959   +173     
- Misses        583      586     +3     

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[rflamary]

thanks @eloitanguy this is very nice work as usual. I have a couple comments below

[rflamary]

could you accept a callable (if same cost for eveyone) and a list?

[rflamary]

not a fan of teh exception, because of the try there nuerical error in teh loop might not be detected... could you do a classical keak with variable stopingcriterion please?

[rflamary]

cold you please do a function fr that (torch only) in pot that can be used if B is not passed (deafult=None). This will simplify the call for torch cost and keep the possibility to compute closed form.

[rflamary]

Note that it would be nice to have B accept a warm start for X0 also

[rflamary]

could we have different colors for te siferent soure distribs ()and maybe black/gray for the barycenter?

[eloitanguy]

After discussion with @rflamary we decided to expand the PR with additional features, namely two barycenter solvers:

• a solver for the iterates of H of 76 with an optional ground_bary function computing barycenters between the points on different spaces, and with a user-defined notion of cost on each space $\mathcal{Y}_k$ with respect to which the barycentric projection is performed (defaults to L2, which is fast but often not accurate/meaningful depending on the spaces and costs)
• another BCD solver which alternates between the plans and the the positions X (optimisation w.r.t. the positions with a pytorch optimiser)