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We show non-asymptotic exponential convergence of Sinkhorn iterates to theSchr\"odinger potentials, solutions of the quadratic Entropic Optimal Transportproblem on $\mathbb{R}^d$. Our results hold under mild assumptions on themarginal inputs: in particular, we only assume that they admit anasymptotically positive log-concavity profile, covering as special caseslog-concave distributions and bounded smooth perturbations of quadraticpotentials. More precisely, we provide exponential $\mathrm{L}^1$ and pointwiseconvergence of the iterates (resp. their gradient and Hessian) to Schr\"odingerpotentials (resp. their gradient and Hessian) for large enough values of theregularization parameter. As a corollary, we establish exponential convergenceof Sinkhorn plans with respect to a symmetric relative entropy and inWasserstein distance of order one. Up to the authors' knowledge, these are thefirst results which establish exponential convergence of Sinkhorn's algorithmin a general setting without assuming bounded cost functions or compactlysupported marginals.

Quantitative contraction rates for Sinkhorn algorithm: beyond bounded costs and compact marginals