Sharp local sparsity of regularized optimal transport
This provides theoretical insights into sparsity in regularized optimal transport, generalizing prior results to multivariate cases beyond self-transport, which is incremental for the mathematical optimization community.
The paper tackles the convergence rate of entropy-regularized optimal transport solutions as regularization vanishes, proving that conditional measure supports shrink like balls of radius ε^(1/(d(p-1)+2)) away from boundaries, leading to uniform strong convexity of potentials and convergence rates.
In recent years, the use of entropy-regularized optimal transport with $L^p$-type entropies has become increasingly popular. In this setting, the solutions are sparse, in the sense that the support of the regularized optimal coupling, $\mathrm{supp}(Ï_\varepsilon)$, shrinks to the support of the original optimal transport problem as $\varepsilon \to 0$. The main open question concerns the rate of this convergence. In this paper, we obtain sharp local results away from the boundary. We prove that the supports $\mathrm{supp}(Ï_\varepsilon(\cdot \mid x))$ of the conditional measures, $Ï_\varepsilon(\cdot \mid x)$, behave like balls of radius $\varepsilon^\frac 1 {d(p-1)+2}$. This allows us to show that the regularized potentials are uniformly strongly convex and to derive the rate of convergence of these potentials toward their unregularized limit. Our results generalize the results of (González-Sanz and Nutz, SIAM J.~Math.~Anal.) and (Wiesel and Xu, Ibid.) to the multivariate case and beyond the case of self-transport.