[WIP] Bures-Wasserstein Gradient Descent for Bures-Wasserstein Barycenters

[clbonet]

Types of changes

This PR aims to add the Bures-Wasserstein gradient descent solver to compute Bures-Wasserstein barycenters (see e.g. Gradient descent algorithms for Bures-Wasserstein barycenters or Averaging on the Bures-Wasserstein manifold: dimension-free convergence of gradient descent).

• Restructured ot.gaussian.bures_wasserstein_barycenter to allow to use different methods
• Added the previous fixed-point algorithm in ot.gaussian.bures_barycenter_fixpoint
• Added the Bures-Wasserstein gradient descent in ot.gaussian.bures_barycenter_gradient_descent
• Added an iteration over the methods in the test test_bures_wasserstein_barycenter
• Added a test test_fixedpoint_vs_gradientdescent_bures_wasserstein_barycenter
• Added batch version of ot.gaussian.bures_wasserstein_distance
• Trace can be computed for batchs of matrices

Motivation and context / Related issue

The Bures-Wasserstein gradient descent comes with convergence guarantees to solve Bures-Wasserstein barycenters. Moreover, it can also be used in a stochastic way when there are too much Gaussian. Thus, it is a good alternative to the fixed-point algorithm currently implemented.

How has this been tested (if it applies)

I added a test test_fixedpoint_vs_gradientdescent_bures_wasserstein_barycenter to assess both methods returns the same barycenter. I also added the itertools to test_bures_wasserstein_barycenter.

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.

[rflamary]

Small comments. I will let @antoinecollas do a proper review he is the expert in Riemannian optimization

[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 97.09%. Comparing base (1761d0b) to head (f669a8e).

Additional details and impacted files

@@            Coverage Diff             @@
##           master     #680      +/-   ##
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+ Coverage   97.05%   97.09%   +0.04%     
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  Files          98       98              
  Lines       19955    20167     +212     
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+ Hits        19367    19582     +215     
+ Misses        588      585       -3     

[rflamary]

This is great. A few tests especialy about errors are missing

[rflamary]

not tested