Sparse Regularized Optimal Transport without Curse of Dimensionality

TL;DR

This work establishes that sparse, regularized optimal transport regularized by broad f-divergences attains parametric-rate convergence in the regime of i.i.d. marginals, independent of ambient dimension. By developing a Z-estimation-based analytical framework and leveraging Hölder-space techniques, it proves central limit theorems for the optimal cost, the empirical coupling, and the dual potentials even when the regularization induces limited smoothness and sparsity. A key contribution is the invertibility of the linearized first-order condition around population potentials, enabling a three-term decomposition that bypasses the need for Donsker properties. The results provide precise statistical guarantees for ROT and pave the way for robust inference in high-dimensional, regularized OT applications where sparsity is desirable. Practically, this advances understanding of how sparse regularized couplings behave under sampling, with implications for statistical OT, barycenter estimation, and related inference tasks.

Abstract

Entropic optimal transport -- the optimal transport problem regularized by KL diver\-gence -- is highly successful in statistical applications. Thanks to the smoothness of the entropic coupling, its sample complexity avoids the curse of dimensionality suffered by unregularized optimal transport. The flip side of smoothness is overspreading: the entropic coupling always has full support, whereas the unregularized coupling that it approximates is usually sparse, even given by a map. Regularizing optimal transport by less-smooth $f$-divergences such as Tsallis divergence (i.e., $L^p$-regularization) is known to allow for sparse approximations, but is often thought to suffer from the curse of dimensionality as the couplings have limited differentiability and the dual is not strongly concave. We refute this conventional wisdom and show, for a broad family of divergences, that the key empirical quantities converge at the parametric rate, independently of the dimension. More precisely, we provide central limit theorems for the optimal cost, the optimal coupling, and the dual potentials induced by i.i.d.\ samples from the marginals. These results are obtained by a powerful yet elementary approach that is of broader interest for Z-estimation in function classes that are not Donsker.

Figures (1)

• Figure 1: Log-log plots of $\left|\mathrm{OT}_ \varepsilon(P_n, Q_n) - \mathrm{OT}_ \varepsilon(P, Q)\right|$ for dimensions $d \in \{1, 5, 10\}$ (left, middle, right) and two different divergences, namely $\varphi(x) = \frac{2}{3}(x^{3/2} -1)$ leading to $\psi(x) = \frac{1}{3} x_+^3 + \frac{2}{3}$ (top) and $\varphi(x) = \frac{1}{2} (x^2 - 1)$ leading to $\psi(x) = \frac{1}{2} x_+^2 + \frac{1}{2}$ (bottom). We use $\varepsilon=0.5$. We numerically observe approximate rates $n^{\alpha}$ with $\alpha \approx -1$ for all cases, irrespective of dimension. The approximation is based on thirty independent evaluations of $\left|\mathrm{OT}_ \varepsilon(P_n, Q_n) - \mathrm{OT}_ \varepsilon(P, Q)\right|$ illustrated by the gray dots, while the black dots are the respective averages for each $n\in\{10, 30, 100, 300, 1000, 3000\}$.

Theorems & Definitions (41)

• Example 2.2
• Proposition 2.3
• Remark 2.4: Limited Smoothness
• Remark 2.5: Regularization parameter
• Theorem 3.2
• Theorem 3.4: CLT for potentials
• Theorem 3.5: CLT for costs
• Theorem 3.6: CLT for couplings
• Remark 3.7: Regularization parameter
• Lemma 4.1