An exact and fast solution of the inverse Regularized Optimal Transport problem
arXiv:2609.01278
2026
Architecture
2 ideas extracted · analyzed Sep 2, 2026
What the math gives to ML
The paper provides an exact inverse map for entropic optimal transport: from an observed coupling W, the cost matrix can be recovered up to the unavoidable row-plus-column gauge by applying double-centering to log W. This removes the Sinkhorn scaling potentials without iterative optimization, and the same structure permits estimating the entropic temperature when some true cost entries are known. The strongest neural-network transfer is a Sinkhorn-constrained attention or routing layer whose observed transport plan is converted into a gauge-fixed cost matrix for initialization, distillation, monitoring, or adaptive temperature control. The key falsifiable signature is exact affine recovery of centered costs, with degradation only when entries of W approach numerical zero.
Ideas from this paper
✗ Failed on benchmark
2026
Replace ordinary row-softmax attention with a doubly stochastic Sinkhorn attention plan W, and periodically recover a gauge-fixed pairwise cost from W using the exact double-centering identity. Use this recovered cost to initialize or regularize a structured attention score, making the attention geometry identifiable despite arbitrary query and key row and column offsets.
Useful7/10
Difficulty5/10
Novelty7/10
Unverified
2026
Use a small set of known or trusted pairwise costs to estimate the effective entropic temperature of a Sinkhorn attention or mixture-of-experts routing layer directly from its observed transport plan. This provides a calibration controller that can detect over-concentrated routing and adjust epsilon without backpropagating through a costly temperature search.
Useful6/10
Difficulty4/10
Novelty8/10