Entropic Transport Coupling Bias

Entropic optimal transport modifies the classical optimal transport problem by adding a Kullback-Leibler divergence penalty against a reference independence coupling. This regularization makes the problem strictly convex and efficiently solvable via matrix scaling (Sinkhorn's algorithm), but it introduces a bias that shifts the optimal transport plan away from the unregularized minimum-cost coupling toward the independence coupling.

Experimenting with Regularization

You can interact with the Entropic Transport Coupling Bias module (https://riano.app/module/entropic-transport-coupling-bias) to observe how different settings affect the coupling:

1. Set Cost matrix to "Diagonal free" and Cost gap to 1.
2. Change Regularization epsilon from 0 to 2. Notice how the transport cost increases from 0 (at epsilon 0) as the coupling is drawn toward the uniform independent distribution.
3. Change Cost matrix to "Cross free". The plan matrix swaps its diagonal and off-diagonal entries, matching the new cost structure while preserving the exact same regularized objective value.
4. Set Cost matrix to "Flat". Observe that the transport cost equals the cost gap regardless of epsilon, and at epsilon=0 the minimizer becomes nonunique.