Shayan Hundrieser

Selected Publications

1. Minimax Optimality of the NPMLE for Gaussian Mixtures in Hellinger Distance

   Shayan Hundrieser, and Tudor Manole

   Preprint https://arxiv.org/abs/2610.30109 - 2026

   Abstract Bibtex Link

   We analyze the problem of estimating univariate Gaussian mixture densities using the nonparametric maximum likelihood estimator (NPMLE) based on n independent and identically distributed samples. We show that the unconstrained NPMLE achieves the minimax optimal rate log(n)/n in squared Hellinger distance, when the true mixing measure is sub-Gaussian. In addition, we show that when the true mixing measure has compact support, the NMPLE constrained to a compact interval also achieves the minimax optimal rate, which is slightly faster and of order log(n)/(nloglogn). Our proof consists of sharply characterizing the local Hellinger bracketing entropy of the class of finite Gaussian mixtures. One of our key technical innovations is to sharply bound the L2 bracketing entropy of balls of bounded-degree polynomials, which may be of independent interest. As an application, we establish the regret-optimality of g-modeling procedures based on the NPMLE in the Gaussian Empirical Bayes problem with bounded priors.

   @article{Hundrieser2026NPMLE,
     title = {Minimax Optimality of the NPMLE for Gaussian Mixtures in Hellinger Distance},
     author = {Hundrieser, Shayan and Manole, Tudor},
     year = {2026},
   }

2. Hyper Input Convex Neural Networks for Shape Constrained Learning and Optimal Transport

   Shayan Hundrieser, Insung Kong, and Johannes Schmidt-Hieber

   NeuRIPS [preprint arXiv:2604.26942] - 2026

   Abstract Bibtex Link

   We introduce Hyper Input Convex Neural Networks (HyCNNs), a novel neural network architecture designed for learning convex functions. HyCNNs combine the principles of Maxout networks with input convex neural networks (ICNNs) to create a neural network that is always convex in the input, theoretically capable of leveraging depth, and performs reliable when trained at scale compared to ICNNs. Concretely, we prove that HyCNNs require exponentially fewer parameters than ICNNs to approximate quadratic functions up to a given precision. Throughout a series of synthetic experiments, we demonstrate that HyCNNs outperform existing ICNNs and MLPs in terms of predictive performance for convex regression and interpolation tasks. We further apply HyCNNs to learn high-dimensional optimal transport maps for synthetic examples and for single-cell RNA sequencing data, where they oftentimes outperform ICNN-based neural optimal transport methods and other baselines across a wide range of settings.

   @article{Hundrieser2026HyCNN,
     title = {Hyper Input Convex Neural Networks for Shape Constrained Learning and Optimal Transport},
     author = {Hundrieser, Shayan and Kong, Insung and Schmidt-Hieber, Johannes},
     journal = {NeuRIPS [preprint arXiv:2604.26942]},
     year = {2026},
   }

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},
   }