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1. Projections of orbital measures and quantum marginal problems

   https://doi.org/10.1090/tran/8931  

   Collins, Benoît; McSwiggen, Colin (August 2023, Transactions of the American Mathematical Society)

   This paper studies projections of uniform random elements of (co)adjoint orbits of compact Lie groups. Such projections generalize several widely studied ensembles in random matrix theory, including the randomized Horn’s problem, the randomized Schur’s problem, and the orbital corners process. In this general setting, we prove integral formulae for the probability densities, establish some properties of the densities, and discuss connections to multiplicity problems in representation theory as well as to known results in the symplectic geometry literature. As applications, we show a number of results on marginal problems in quantum information theory and also prove an integral formula for restriction multiplicities. 

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2. COMET Flows: Towards Generative Modeling of Multivariate Extremes and Tail Dependence

   https://doi.org/10.24963/ijcai.2022/462  

   McDonald, Andrew; Tan, Pang-Ning; Luo, Lifeng (July 2022, Proceedings of the Thirty-First International Joint Conference on Artificial Intelligence)

   Normalizing flows—a popular class of deep generative models—often fail to represent extreme phenomena observed in real-world processes. In particular, existing normalizing flow architectures struggle to model multivariate extremes, characterized by heavy-tailed marginal distributions and asymmetric tail dependence among variables. In light of this shortcoming, we propose COMET (COpula Multivariate ExTreme) Flows, which decompose the process of modeling a joint distribution into two parts: (i) modeling its marginal distributions, and (ii) modeling its copula distribution. COMET Flows capture heavy-tailed marginal distributions by combining a parametric tail belief at extreme quantiles of the marginals with an empirical kernel density function at mid-quantiles. In addition, COMET Flows capture asymmetric tail dependence among multivariate extremes by viewing such dependence as inducing a low-dimensional manifold structure in feature space. Experimental results on both synthetic and real-world datasets demonstrate the effectiveness of COMET flows in capturing both heavy-tailed marginals and asymmetric tail dependence compared to other state-of-the-art baseline architectures. All code is available at https://github.com/andrewmcdonald27/COMETFlows. 

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3. Recovering Joint PMF from Pairwise Marginals

   https://doi.org/10.1109/IEEECONF51394.2020.9443425  

   Ibrahim, Shahana; Fu, Xiao (November 2020, Asilomar Conference on Signals, Systems, and Computers)

   null (Ed.)

   To overcome the curse of dimensionality in joint probability learning, recent work has proposed to recover the joint probability mass function (PMF) of an arbitrary number of random variables (RVs) from three-dimensional marginals, exploiting the uniqueness of tensor decomposition and the (unknown) dependence among the RVs. Nonetheless, accurately estimating three-dimensional marginals is still costly in terms of sample complexity. Tensor decomposition also poses a computationally intensive optimization problem. This work puts forth a new framework that learns the joint PMF using pairwise marginals that are relatively easy to acquire. The method is built upon nonnegative matrix factorization (NMF) theory, and features a Gram–Schmidt-like economical algorithm that works provably well under realistic conditions. Theoretical analysis of a recently proposed expectation maximization (EM) algorithm for joint PMF recovery is also presented. In particular, the EM algorithm is shown to provably improve upon the proposed pairwise marginal-based approach. Synthetic and real-data experiments are employed to showcase the effectiveness of the proposed approach. 

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4. Inferring Evanescent Stochastic Dynamics from Marginal Distributions

   https://doi.org/10.1007/s11424-025-4440-9  

   Eldesoukey, Asmaa; Miangolarra, Olga Movilla; Georgiou, Tryphon T (February 2025, Journal of Systems Science and Complexity)

   Evanescent random walks are instances of stochastic processes that terminate at a specific rate. They have proved relevant in modeling diverse behaviors of complex systems from protein degra- dation in gene networks (Ali and Brewster (2022), Ghusinga, et al. (2017), and Ham, et al. (2024)) and “nonprocessive” motor proteins (Kolomeisky and Fisher (2007)) to decay of diffusive radioactive matter (Zoia (2008)). The present work aims to extend a well-established estimation and control problem, the so-called Schr¨odinger’s bridge problem, to evanescent diffusion processes. Specifically, the authors seek the most likely law on the path space that restores consistency with two marginal densities — One is the initial probability density of the flow, and the other is a density of killed particles. The Schr¨odinger’s bridge problem can be interpreted as an estimation problem but also as a control problem to steer the stochastic particles so as to match specified marginals. The focus of previous work in Eldesoukey, et al. (2024) has been to tackle Schr¨odinger’s problem involving a constraint on the spatio-temporal density of killed particles, which the authors revisit here. The authors then expand on two related problems that instead separately constrain the temporal and the spatial marginal densities of killed particles. The authors derive corresponding Schr¨odinger systems that contain coupled partial differential equa- tions that solve such problems. The authors also discuss Fortet-Sinkhorn-like algorithms that can be used to construct the sought bridges numerically. 

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5. Particle approximations of Wigner distributions for $n$ arbitrary observables

   https://doi.org/10.1103/PhysRevResearch.7.013102  

   Sabbagh, Ralph; Movilla_Miangolarra, Olga; Hezari, Hamid; Georgiou, Tryphon T (January 2025, Physical Review Research)

   A class of signed joint probability measures for $n$arbitrary quantum observables is derived and studied based on quasicharacteristic functions with symmetrized operator orderings of Margenau-Hill type. It is shown that the Wigner distribution associated with these observables can be rigorously approximated by such measures. These measures are given by affine combinations of Dirac delta distributions supported over the finite spectral range of the quantum observables and give the correct probability marginals when coarse-grained along any principal axis. We specialize to bivariate quasiprobability distributions for the spin measurements of spin- $1/2$particles and derive their closed-form expressions. As a side result, we point out a connection between the convergence of these particle approximations and the Mehler-Heine theorem. Finally, we interpret the supports of these quasiprobability distributions in terms of repeated thought experiments. Published by the American Physical Society2025 

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