[MRG] Fix sinkhorn and stabilized barycenters with non-uniform weights

[itzzdev09]

Types of changes

• Bug fix

Description

With non-uniform weights, ot.bregman.barycenter gives different results depending on method, and only "sinkhorn_log" is correct:

A = np.stack([ot.datasets.make_1D_gauss(15, 4, 2), ot.datasets.make_1D_gauss(15, 10, 2)], 1)
M = ot.utils.dist0(15); M /= M.max()
w = np.array([0.3, 0.7])
lg = ot.bregman.barycenter(A, M, 1e-2, w, method="sinkhorn_log")
np.abs(ot.bregman.barycenter(A, M, 1e-2, w, method="sinkhorn") - lg).max()              # 1.2e-2
np.abs(ot.bregman.barycenter(A, M, 1e-2, w, method="sinkhorn_stabilized") - lg).max()   # 1.8e-1
ot.bregman.barycenter(A, M, 1e-3, np.array([0.5, 0.5]), method="sinkhorn_stabilized").sum()  # 0.599

Evaluating the objective sum_k w_k KL(pi_k | K) confirms that the "sinkhorn_log" result is the minimum (e.g. 33.056 against 33.169 for "sinkhorn"). Two separate problems:

• barycenter_sinkhorn: the update u <- u * geometricBar(weights, UKv) / UKv leaves the weighted geometric mean of u unchanged, so the barycenter is only optimal if it starts at 1. u was initialised with geometricMean (uniform weights) instead of geometricBar(weights, ...), so with non-uniform weights the iterations converge to a different fixed point. This is a one-line change in the initialisation.
• barycenter_stabilized: when absorbing, the common scaling max_k v_k is moved into K, but v was then reset to ones for every histogram, which is only right when all histograms have the same v. Also, after an absorption K.T @ u includes exp(beta / reg), which was not taken out of the geometric mean, so q was off by that factor. Now v is divided by the absorbed max, and q is computed with the - beta / reg correction.

After the change the three methods agree to ~1e-11 for all the weights and reg values I tried, and the stabilized barycenter sums to one. The existing test_barycenter only uses alpha = 0.5, and at reg=1e-2 with equal weights the stabilized version happened to be right, which is why these went unnoticed.

How has this been tested

Added test_barycenter_non_uniform_weights (reg 1e-3 and 1e-2, weights [0.2, 0.8]), which checks that all three methods agree and sum to one. It fails on master. test_bregman.py, test/unbalanced, test_da.py and test_utils.py pass.

Checklist

• I have added tests to cover my changes.
• I have updated the RELEASES.md file.

[codecov]

Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 96.88%. Comparing base (98d09a1) to head (375eefb).

Additional details and impacted files

@@            Coverage Diff             @@
##           master     #870      +/-   ##
==========================================
+ Coverage   96.86%   96.88%   +0.02%     
==========================================
  Files         128      128              
  Lines       26304    26319      +15     
==========================================
+ Hits        25480    25500      +20     
+ Misses        824      819       -5     

🚀 New features to boost your workflow:

• ❄️ Test Analytics: Detect flaky tests, report on failures, and find test suite problems.