Related papers: Stability of Quadratically Regularized Optimal Tra…
We study the quadratically regularized optimal transport (QOT) problem for quadratic cost and compactly supported marginals $\mu$ and $\nu$. It has been empirically observed that the optimal coupling $\pi_\epsilon$ for the QOT problem has…
We study the stability of entropically regularized optimal transport with respect to the marginals. Lipschitz continuity of the value and H\"older continuity of the optimal coupling in $p$-Wasserstein distance are obtained under general…
Quadratically regularized optimal transport (QOT) is an alternative to entropic regularization that yields sparse couplings and avoids numerical instabilities due to exponential scaling. From an optimization viewpoint, the dual QOT…
It is well known that optimal transport suffers from the curse of dimensionality: when the prescribed marginals are approximated by i.i.d. samples, the convergence of the empirical optimal transport problem to the population counterpart…
This work investigates several aspects related to quantitative stability in optimal transport, as well as uniqueness of the dual transport problem. Our main contributions are as follows. Chapter 1: Observations regarding the quantitative…
Quadratic regularization has emerged as a potential alternative to the popular entropic regularization in computational optimal transport, offering the theoretical advantage of producing sparse couplings through its hinge density structure.…
We consider regularised quadratic optimal transport with subquadratic polynomial or entropic regularisation. In both cases, we prove interior Lipschitz-estimates on a transport-like map and interior gradient Lipschitz-estimates on the…
In this note, we derive upper-bounds on the statistical estimation rates of unbalanced optimal transport (UOT) maps for the quadratic cost. Our work relies on the stability of the semi-dual formulation of optimal transport (OT) extended to…
We introduce the framework of quadratic-form optimal transport (QOT), whose transport cost has the form $\iint c\,\mathrm{d}\pi \otimes\mathrm{d}\pi$ for some coupling $\pi$ between two marginals. Interesting examples of quadratic-form…
We establish novel quantitative stability results for optimal transport problems with respect to perturbations in the target measure. We provide explicit bounds on the stability of optimal transport potentials and maps, which are relevant…
The optimal transport problem with quadratic regularization is useful when sparse couplings are desired. The density of the optimal coupling is described by two functions called potentials; equivalently, potentials can be defined as a…
The quadratically regularized optimal transport problem has recently been considered in various applications where the coupling needs to be \emph{sparse}, i.e., the density of the coupling needs to be zero for a large subset of the product…
We study stability of optimizers and convergence of Sinkhorn's algorithm for the entropic optimal transport problem. In the special case of the quadratic cost, our stability bounds imply that if one of the two entropic potentials is…
Partial Optimal Transport (POT) has recently emerged as a central tool in various Machine Learning (ML) applications. It lifts the stringent assumption of the conventional Optimal Transport (OT) that input measures are of equal masses,…
We study stability and sample complexity properties of divergence regularized optimal transport (DOT). First, we obtain quantitative stability results for optimizers of DOT measured in Wasserstein distance, which are applicable to a wide…
In optimal transport, quadratic regularization is a sparse alternative to entropic regularization: the solution measure tends to have small support. Computational experience suggests that the support decreases monotonically to the…
We study the quantitative stability of the mapping that to a measure associates its pushforward measure by a fixed (non-smooth) optimal transport map. We exhibit a tight H\"older-behavior for this operation under minimal assumptions. Our…
We prove quantitative bounds on the stability of optimal transport maps and Kantorovich potentials from a fixed source measure $\rho$ under variations of the target measure $\mu$, when the cost function is the squared Riemannian distance on…
We derive nearly tight and non-asymptotic convergence bounds for solutions of entropic semi-discrete optimal transport. These bounds quantify the stability of the dual solutions of the regularized problem (sometimes called Sinkhorn…
We establish several quantitative stability estimates for optimal transport maps between non-degenerate densities on uniformly convex domains for the quadratic cost. Under H\"older regularity assumptions, we prove Lipschitz $L^2$…