Uniform Statistical Convergence of Empirical Sinkhorn Potentials with Exponential and Polynomial Dependence on the Regularization Parameter

What the math gives to ML

The paper isolates a useful decomposition for entropic transport estimation: empirical operator fluctuations scale as n^(-1/2), while inverse stability of the population Sinkhorn fixed point controls how strongly those fluctuations are amplified. The transferable asset is the explicit polynomial dependence on the entropic temperature ε, which can prevent neural transport modules from selecting an unnecessarily sharp and unstable temperature. A practical adaptation is a Sinkhorn attention or MoE router whose ε is adjusted using online estimates of residual sensitivity and minibatch complexity rather than treated as a fixed hyperparameter. This creates a falsifiable tradeoff between sharper assignments and routing noise.