[MRG] 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
• added a solver for the "true" fixed-point algo which increases support sizes across iterations

For the (theoretical) fixed-point method, use method='true_fixed_point' in ot.lp.free_support_barycenter_generic_costs and for the barycentric heuristic, use method='L2_barycentric_proj'. The latter is the default, given the computational advantages and the desirable property of keeping a fixed support size.

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 99.24812% with 3 lines in your changes missing coverage. Please review.

Project coverage is 97.13%. Comparing base (99477be) to head (9a344e1).

Additional details and impacted files

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##           master     #715      +/-   ##
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+ Coverage   97.09%   97.13%   +0.04%     
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  Files         100      100              
  Lines       20432    20828     +396     
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+ Hits        19839    20232     +393     
- Misses        593      596       +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?

[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)

[eloitanguy]

After some updates to the paper behind these algorithms (the paper update is soon to come), I implemented another barycenter solver which corresponds exactly to the method studied theoretically in the paper (iterates of G). I updated this PR with the additional algorithm (method='true_fixed_point'), with a comparison of the two methods in the example.

On my end, this contribution is ready for review :D