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1. Convex relaxation for Fokker–Planck equation

   https://doi.org/10.1098/rspa.2024.0001  

   Chen, Yian; Khoo, Yuehaw; Lim, Lek-Heng (May 2025, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences)

   We propose an approach to directly estimate the moments or marginals for a high-dimensional equilibrium distribution in statistical mechanics by solving the high-dimensional Fokker–Planck equation in terms of low-order cluster moments or marginals. With this approach, we bypass the exponential complexity of estimating the full high-dimensional distribution and directly solve the simplified partial differential equations for low-order moments/marginals. Moreover, the proposed moment/marginal relaxation is fully convex and can be solved via off-the-shelf solvers. We further propose a time-dependent version of the convex programs to study non-equilibrium dynamics. In a specific setting, we show the proposed method can recover a mean-field-type equilibrium density. Numerical results are provided to demonstrate the performance of the proposed algorithm for high-dimensional systems. 

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2. 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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3. 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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4. 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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5. Optimal Survival Strategies for Diffusive Flows: A Schrödinger Bridge Approach to Unbalanced Transport

   https://doi.org/10.1137/25M176581X  

   Chen, Yongxin; Georgiou, Tryphon T; Pavon, Michele (August 2025, SIAM Review)

   Diffusive flows, and their discrete counterparts, are ubiquitous in the physical and engineer- ing sciences. In many important examples, the total mass is not preserved and therefore standard probabilistic models are not suitable. Examples include electrons which may be absorbed by the medium in which they travel. In population genetics, some individuals may ``disappear"" due to their genotype. In traffic flows over a network, some vehicles might simply exit the circulation and park. In this more general situation, where some of the mass may be lost, it is of particular interest to reconcile the observed initial and final marginal distributions with a given prior. In the case when the two marginals are proba- bility distributions, and thus of equal mass, this problem was posed and, to a considerable extent, solved by E. Schr\"odinger in 1931/32. It is now known as the Schr\"odinger Bridge Problem (SBP). It turns out that Schr\"odinger's problem can be viewed as both a modeling and a control problem. Due to the fundamental significance of this problem, interest in the SBP and in its deterministic (zero-noise limit) counterpart of optimal mass transport (OMT) has in recent years enticed scientists from a broad spectrum of disciplines, includ- ing physics, stochastic control, computer science, probability theory, and geometry. Yet, while the mathematics and applications of SBP/OMT have been developing at a consid- erable pace, accounting for marginals of unequal mass has received scant attention. The problem of interpolating between ``unbalanced"" marginals has been approached by intro- ducing source/sink terms into the transport equations in an ad hoc manner, chiefly driven by applications in image registration. Nevertheless, as hinted at above, losses are inherent in many physical processes and, thereby, models that account for lossy transport may also need to be reconciled with observed marginals following Schr\"odinger's quest, that is, to adjust the probability of trajectories of particles, including those that do not make it to the terminal observation point, so that the updated evolution represents the most likely way that particles may have been transported, or vanished, at some intermediate point. Thus, the purpose of this work is to develop such a natural generalization of the SBP for diffusive evolution with losses, whereupon particles are ``killed"" (jump into a coffin/extinction state) according to a probabilistic law, and thereby mass is gradually lost along their stochasti- cally driven flow. Through a suitable embedding, which appears to be novel, we turn the problem into an SBP for stochastic processes that combine diffusive and jump character- istics. Then, following a large-deviations formalism in the style of E. Schr\"odinger, given a prior law that allows for losses, we ask for the most probable evolution of particles along with the most likely killing rate as the particles transition between the specified marginals. Our approach differs sharply from previous work involving a Feynman--Kac multiplicative reweighing of the reference measure. The latter, as we argue, is far from Schr\"odinger's quest. An iterative scheme, generalizing the celebrated Fortet--IPF--Sinkhorn algorithm, permits the computation of the new drift and the new killing rate of the path-space solution measure. We also formulate and solve a related fluid-dynamic control problem for the flow of one-time marginals where both the drift and the new killing rate play the role of control variable. A numerical example illustrating the new theoretical results is also presented. 

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