[MRG] Add backend-agnostic semi-discrete OT module and SGD-based solver in ot.semidiscrete

[Ferdinand-Genans]

Types of changes

• New feature (non-breaking change which adds functionality)
• Bug fix (non-breaking change which fixes an issue)
• Breaking change (fix or feature that would cause existing functionality to change)
• Documentation / Examples update

Motivation and context / Related issue

Semi-discrete optimal transport — continuous source, discrete target — is a setting that does not naturally fit POT's discrete-discrete solvers, yet appears in many applications: Monge–Kantorovich statistical depth and quantiles, generative modeling with a continuous prior, Brenier-map estimation, and any setup where the source is given by a sampler rather than a finite empirical distribution.

This PR adds ot.semidiscrete, a single-file backend-agnostic solver for the semi-dual of semi-discrete OT. Every public function takes a reg: float argument, so the user can switch between unregularized OT (reg=0) and the entropic formulation (reg>0) with a single parameter. The underlying algorithm is Averaged SGD on the semi-dual; optionally, the regularization can be decreased throughout the iterations via the DRAG schedule ([83] in README.md; Genans et al., NeurIPS 2025), which empirically improves convergence, especially on large scale problems, when the target regularization is small.

No existing issue tracks this addition.

How has this been tested

• test/test_semidiscrete.py — 34 tests, automatically parametrized over every available POT backend (NumPy and PyTorch on the CI runner). Covers:
  • convergence to a known closed-form optimum on three toy problems (regular grid, nonuniform target weights, shifted 1D);
  • both plain SGD and the DRAG schedule;
  • the entropic regime;
  • a user-supplied custom cost;
  • helper functions (atom_weights row-stochasticity, c_transform reducing to min_j c(x, y_j) at g = 0, ot_map shape and finiteness);
  • solver options (warm start, projection bound, log dict, polyak_average=False, init_potential is not mutated).
• One doctest on solve_semidiscrete (a 500-iter smoke run).
• Full suite + doctest run locally: 35 passed in 5.88 s.
• examples/others/plot_semidiscrete.py runs end-to-end in ~5 s on CPU NumPy and produces the Laguerre-cells figure that becomes the gallery page.
• Also verified locally a larger scale problem on GPU with PyTorch: discrete measure with 20 000 atoms and 30 000 iterations of minibatch of 64 run in ~17 s on a single CUDA device.

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

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 96.84%. Comparing base (60cbe12) to head (6bfe7b8).

Additional details and impacted files

@@            Coverage Diff             @@
##           master     #812      +/-   ##
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+ Coverage   96.81%   96.84%   +0.03%     
==========================================
  Files         126      128       +2     
  Lines       25104    25363     +259     
==========================================
+ Hits        24305    24564     +259     
  Misses        799      799              

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[rflamary]

Hello @Ferdinand-Genans ,

thanks for the PR, we have since merged a paper, could you please resolve conflits and update the reference number ()there is already a 83 now in teh readme file).

I will have a look and do a proper code review ASAP.

[Ferdinand-Genans]

Hi @rflamary,
I think it is good now.
Best regards

[rflamary]

Hello @Ferdinand-Genans ,

This is a very nice PR, I did a code reveiw and have a few comments mainly on the API to make i more consistent with the other POT functions. Also some suggetsion for the example.