Convergence of the Sinkhorn algorithm when the Schrödinger problem has no solution
Bibliographic record
Abstract
The Sinkhorn algorithm is one of the most popular methods for solving the Schrödinger problem: it is known to converge as soon as the latter has a solution, and with a linear rate when the solution has the same support as the reference coupling. Motivated by recent applications of the Schrödinger problem where structured stochastic processes lead to degenerate situations with possibly no solution, we show that the Sinkhorn algorithm still gives rise in this case to exactly two limit points, that can be used to compute the solution of a relaxed version of the Schrödinger problem, which appears as the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Γ</mml:mi> </mml:math> -limit of a problem where the marginal constraints are replaced by marginal penalizations. These results also allow to develop a theoretical procedure for characterizing the support of the solution – both in the original and in the relaxed problem – for any reference coupling and marginal constraints. We showcase promising numerical applications related to a model used in cell biology.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.003 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.006 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".