Shayan Hundrieser

Selected Publications

1. Lower Complexity Adaptation for Empirical Entropic Optimal Transport

   Michel Groppe, and Shayan Hundrieser

   Preprint arXiv:2306.13580 - 2023

   Abstract Bibtex Link

   Entropic optimal transport (EOT) presents an effective and computationally viable alternative to unregularized optimal transport (OT), offering diverse applications for large-scale data analysis. In this work, we derive novel statistical bounds for empirical plug-in estimators of the EOT cost and show that their statistical performance in the entropy regularization parameter ϵ and the sample size n only depends on the simpler of the two probability measures. For instance, under sufficiently smooth costs this yields the parametric rate n^−1/2 with factor ε^−d/2, where d is the minimum dimension of the two population measures. This confirms that empirical EOT also adheres to the lower complexity adaptation principle, a hallmark feature only recently identified for unregularized OT. As a consequence of our theory, we show that the empirical entropic Gromov-Wasserstein distance and its unregularized version for measures on Euclidean spaces also obey this principle. Additionally, we comment on computational aspects and complement our findings with Monte Carlo simulations. Our techniques employ empirical process theory and rely on a dual formulation of EOT over a single function class. Crucial to our analysis is the observation that the entropic cost-transformation of a function class does not increase its uniform metric entropy by much.

   @article{Groppe2023LCA_EOT,
     title = {Lower Complexity Adaptation for Empirical Entropic Optimal Transport},
     author = {Groppe, Michel and Hundrieser, Shayan},
     journal = {Preprint arXiv:2306.13580},
     year = {2023},
   }

2. Convergence of Empirical Optimal Transport in Unbounded Settings

   Thomas Staudt, and Shayan Hundrieser

   Preprint arXiv:2306.11499 - 2023

   Abstract Bibtex Link

   In compact settings, the convergence rate of the empirical optimal transport cost to its population value is well understood for a wide class of spaces and cost functions. In unbounded settings, however, hitherto available results require strong assumptions on the ground costs and the concentration of the involved measures. In this work, we pursue a decomposition-based approach to generalize the convergence rates found in compact spaces to unbounded settings under generic moment assumptions that are sharp up to an arbitrarily small ε>0. Hallmark properties of empirical optimal transport on compact spaces, like the recently established adaptation to lower complexity, are shown to carry over to the unbounded case.

   @article{staudt2023convergence,
     title = {Convergence of Empirical Optimal Transport in Unbounded Settings},
     author = {Staudt, Thomas and Hundrieser, Shayan},
     journal = {Preprint arXiv:2306.11499},
     year = {2023},
   }

3. Weak Limits for Empirical Entropic Optimal Transport: Beyond Smooth Costs

   Alberto González-Sanz, and Shayan Hundrieser

   Preprint arXiv:2305.09745 - 2023

   Abstract Bibtex Link

   We establish weak limits for the empirical entropy regularized optimal transport cost, the expectation of the empirical plan and the conditional expectation. Our results require only uniform boundedness of the cost function and no smoothness properties, thus emphasizing the far-reaching regularizing nature of entropy penalization. To derive these results, we employ a novel technique that sidesteps the intricacies linked to empirical process theory and the control of suprema of function classes determined by the cost. Instead, we perform a careful linearization analysis for entropic optimal transport with respect to an empirical L2-norm, which enables a streamlined analysis. As a consequence, our work gives rise to new implications for a multitude of transport-based applications under general costs, including pointwise distributional limits for the empirical entropic optimal transport map estimator, kernel methods as well as regularized colocalization curves. Overall, our research lays the foundation for an expanded framework of statistical inference with empirical entropic optimal transport.

   @article{gonzalez2023weak,
     title = {Weak Limits for Empirical Entropic Optimal Transport: Beyond Smooth Costs},
     author = {Gonz{\'a}lez-Sanz, Alberto and Hundrieser, Shayan},
     journal = {Preprint arXiv:2305.09745},
     year = {2023},
   }