Shayan Hundrieser

Selected Publications

1. Lower Complexity Adaptation for Empirical Entropic Optimal Transport

   Michel Groppe, and Shayan Hundrieser

   Journal of Machine Learning Research - 2024

   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 εand 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{Groppe2024LCA_EOT,
     title = {Lower Complexity Adaptation for Empirical Entropic Optimal Transport},
     author = {Groppe, Michel and Hundrieser, Shayan},
     journal = {Journal of Machine Learning Research},
     volume = {25},
     number = {344},
     pages = {1--55},
     year = {2024},
   }

2. Local Poisson Deconvolution for Discrete Signals

   Shayan Hundrieser, Tudor Manole, Danila Litskevich, and Axel Munk

   Preprint arXiv:2508.00824 - 2025

   Abstract Bibtex Link

   We analyze the statistical problem of recovering an atomic signal, modeled as a discrete uniform distribution µ, from a binned Poisson convolution model. This question is motivated, among others, by super-resolution laser microscopy applications, where precise estimation of µ provides insights into spatial formations of cellular protein assemblies. Our main results quantify the local minimax risk of estimating µ for a broad class of smooth convolution kernels. This local perspective enables us to sharply quantify optimal estimation rates as a function of the clustering structure of the underlying signal. Moreover, our results are expressed under a multiscale loss function, which reveals that different parts of the underlying signal can be recovered at different rates depending on their local geometry. Overall, these results paint an optimistic perspective on the Poisson deconvolution problem, showing that accurate recovery is achievable under a much broader class of signals than suggested by existing global minimax analyses. Beyond Poisson deconvolution, our results also allow us to establish the local minimax rate of parameter estimation in Gaussian mixture models with uniform weights. We apply our methods to experimental super-resolution microscopy data to identify the location and configuration of individual DNA origamis. In addition, we complement our findings with numerical experiments on runtime and statistical recovery that showcase the practical performance of our estimators and their trade-offs.

   @article{hundrieser2025local,
     author = {{Hundrieser}, Shayan and {Manole}, Tudor and {Litskevich}, Danila and {Munk}, Axel},
     title = {{Local Poisson Deconvolution for Discrete Signals}},
     journal = {Preprint arXiv:2508.00824},
     year = {2025},
   }

3. Empirical optimal transport between different measures adapts to lower complexity

   Shayan Hundrieser, Thomas Staudt, and Axel Munk

   Annales de l’Institut Henri Poincaré, Probabilités et Statistiques - 2024

   Abstract Bibtex Link

   The empirical optimal transport (OT) cost between two probability measures from random data is a fundamental quantity in transport based data analysis. In this work, we derive novel guarantees for its convergence rate when the involved measures are different, possibly supported on different spaces. Our central observation is that the statistical performance of the empirical OT cost is determined by the less complex measure, a phenomenon we refer to as lower complexity adaptation of empirical OT. For instance, under Lipschitz ground costs, we find that the empirical OT cost based on n observations converges at least with rate n^(−1/d) to the population quantity if one of the two measures is concentrated on a d-dimensional manifold, while the other can be arbitrary. For semi-concave ground costs, we show that the upper bound for the rate improves to n^(−2/d). Similarly, our theory establishes the general convergence rate n^(−1/2) for semi-discrete OT. All of these results are valid in the two-sample case as well, meaning that the convergence rate is still governed by the simpler of the two measures. On a conceptual level, our findings therefore suggest that the curse of dimensionality only affects the estimation of the OT cost when both measures exhibit a high intrinsic dimension. Our proofs are based on the dual formulation of OT as a maximization over a suitable function class F_c and the observation that the c-transform of F_c under bounded costs has the same uniform metric entropy as F_c itself

   @article{hundrieser2024empirical,
     title = {Empirical optimal transport between different measures adapts to lower complexity},
     author = {Hundrieser, Shayan and Staudt, Thomas and Munk, Axel},
     journal = {Annales de l'Institut Henri Poincar\'{e}, Probabilit\'{e}s et Statistiques},
     volume = {60},
     number = {2},
     pages = {824--846},
     year = {2024},
   }