Scalable Computation of Dynamic Flow Problems via Multimarginal Graph-Structured Optimal Transport

The optimal transport problem has recently developed into a powerful framework for various applications in estimation and control. Many of the recent advances in the theory and application of optimal transport are based on regularizing the problem with an entropy term, which connects it to the Schrödinger bridge problem and thus to stochastic optimal control. Moreover, the entropy regularization makes the otherwise computationally demanding optimal transport problem feasible even for large scale settings. This has led to an accelerated development of optimal transport based methods in a broad range of fields. Many of these applications have an underlying graph structure, for instance, information fusion and tracking problems can be described by trees. In this work we consider multimarginal optimal transport problems with a cost function that decouples according to a tree structure. The entropy regularized multimarginal optimal transport problem can be viewed as a generalization of the Schrödinger bridge problem with the same tree-structure, and by utilizing these connections we extend the computational methods for the classical optimal transport problem in order to solve structured multimarginal optimal transport problems in an efficient manner. In particular, the algorithm requires only matrix-vector multiplications of relatively small dimensions. We show that the multimarginal regularization introduces less diffusion, compared to the commonly used pairwise regularization, and is therefore more suitable for many applications. Numerical examples illustrate this, and we finally apply the proposed framework for the tracking of an ensemble of indistinguishable agents.

In machine learning, optimal transport (OT) theory is extensively utilized to compare probability distributions across various applications, such as graph data represented by node distributions and image data represented by pixel distributions. In practical scenarios, it is often necessary to solve multiple OT problems. Traditionally, these problems are treated independently, with each OT problem being solved sequentially. However, the computational complexity required to solve a single OT problem is already substantial, making the resolution of multiple OT problems even more challenging. Although many applications of fast solutions to OT are based on the premise of a single OT problem with arbitrary distributions, few efforts handle such multiple OT problems with multiple distributions. Therefore, we propose the anchor space OT (ASOT) problem: an approximate OT problem designed for multiple OT problems. This proposal stems from our finding that in many tasks the mass transport tends to be concentrated in a reduced space from the original feature space. By restricting the mass transport to a learned anchor point space, ASOT avoids pairwise instantiations of cost matrices for multiple OT problems and simplifies the problems by canceling insignificant transports. This simplification greatly reduces its computational costs. We then prove the upper bounds of its 1-Wasserstein distance error between the proposed ASOT and the original OT problem under different conditions. Building upon this accomplishment, we propose three methods to learn anchor spaces for reducing the approximation error. Furthermore, our proposed methods present great advantages for handling distributions of different sizes with GPU parallelization.

Finding optimal trajectories for multiple traffic demands in a congested network is a challenging task. Optimal transport theory is a principled approach that has been used successfully to study various transportation problems. Its usage is limited by the lack of principled and flexible ways to incorporate realistic constraints. We propose a principled physics-based approach to impose constraints flexibly in optimal transport problems. Constraints are included in mirror descent dynamics using the D'Alembert-Lagrange principle from classical mechanics. This results in a sparse, local and linear approximation of the feasible set leading in many cases to closed-form updates.

This paper studies on the discrete time dynamic flow model and its application. Time is taken as the main consideration, and based on the discrete time unit. Combined with a single source evacuation algorithm, the discrete-time dynamic network flow model for evacuation can be formulated. Given there is uncertainty in the number of escapees, it is unable to calculate the minimal evacuation time. However establishing a network dynamic flow evacuation model, the problem can be transformed into the maximum dynamic network flow problem. The model can produce lower bounds for the evacuation time without considering any individual behavior, the value can be used to evaluate or analyze the evacuation of the community.

This dissertation develops optimal transport-based methodologies to advance statistical dependency analysis and co-localization studies, with applications in biological imaging. First, we introduce transport dependency, a novel measure grounded in optimal transport theory, that quantifies statistical dependence between random variables in general Polish spaces. Due to its broad applicability, transport dependency allows for independence testing on complex data with diverse internal structures, enabling robust detection of dependencies across a wide range of data types. We then introduce a more specialized framework that combines point process theory with optimal transport to describe dependence and co-localization in dual-channel fluorescence microscopy images. Using a Bayesian approach, we demonstrate how prior information can be seamlessly integrated into the analysis through the cost function, enhancing model flexibility and accuracy. Simulations show that this model enables precise quantification of protein interactions, distinguishing between true statistical dependencies and random co-occurrences. Finally, we present MultiMatch, an advanced tool utilizing multi-marginal optimal unbalanced transport to analyze spatial arrangements in multi-color microscopy images. In particular, MultiMatch aims in detecting specific biological structures while addressing challenges posed by incomplete labeling. Validated on DNA origami nanoruler data, MultiMatch consistently outperforms geometry-agnostic methods and supports multi-channel co-localization through accessible, user-friendly visualization tools.

Optimal transport has a long history in mathematics which was proposed by Gaspard Monge in the eighteenth century (Monge, 1781). However, until recently, advances in optimal transport theory pave the way for its use in the AI community, particularly for formulating deep generative models. In this paper, we provide a comprehensive overview of the literature in the field of deep generative models using optimal transport theory with an aim of providing a systematic review as well as outstanding problems and more importantly, open research opportunities to use the tools from the established optimal transport theory in the deep generative model domain.

In this paper, we investigate a three-dimensions (3D) cellular network that incorporates both unmanned aerial vehicle-base stations (UAV-BSs) and UAV-users (UAV-UEs). In the considered model, UAV-BSs hover in 3D space to serve UAV-UEs that are distributed according to an arbitrary spatial distribution function. We propose an optimal 3D cell partitioning method based on fair resource allocation scheme for which the total amount of data service is maximized. Utilizing the mathematical tool of optimal transport theory, the cell partitioning problem can be modeled as a semi-discrete optimal transport problem which is easy to be solved. To this end, we apply steepest-ascent search algorithm to solve the 3D cell partition optimization problem simplified by the optimal transport theory. The simulation results show that fairness of the proposed cell partitioning algorithm is higher than max-SINR algorithm, and it also guarantees the amount of system data service.

Relatively recently, there has been much activity on two particular generalizations of the classical two-marginal optimal transport problem. The first is the partial transport problem, where the total mass of the two distributions to be coupled may not match, and one is forced to choose submeasures of the constraints for coupling. The other generalization is the multi-marginal transport problem, where there are 3 or more probability distributions to be coupled together in some optimal manner. By combining the above two generalizations we obtain a natural extension: the multi-marginal optimal partial transport problem. In joint work with Brendan Pass (University of Alberta), we have obtained uniqueness of solutions (under hypotheses analogous to the two-marginal partial transport problem given by Figalli) by relating the problem to what we deem the “partial barycenter problem” for finite measures. Interestingly enough, we also observe some significantly different behavior of solutions compared to the two marginal case.

MotivationInferring differential gene regulatory networks (GRNs) between different conditions from gene expression profiles remains a significant challenge. Current GRN inference approaches are limited by either scalability in large networks or accuracy in high-dimensional scenarios. Furthermore, most existing methods require paired samples for comparative GRN analyses.ResultsTo overcome these challenges, we model gene regulation as a distribution transportation problem and propose an efficient and effective method, called double optimal transport (OT), for reconstructing differential GRNs from the perspective of optimal transport theory, applicable to unpaired samples. Double OT is a novel two-level OT framework. It first aligns unpaired samples by solving a partial OT problem at the sample level, and then infers GRNs from the aligned samples by solving a robust OT problem at the gene level. Comprehensive simulation studies demonstrate the superior efficiency and efficacy of double OT in different scales of networks compared to state-of-the-art methods. We also apply the proposed method to a gastric cancer dataset, identifying the proto-oncogene MET as a central node in the gastric cancer GRN. Its crucial role in early oncogenesis and potential as a therapeutic target further validate our approach and enhance our understanding of the regulatory mechanisms of gastric cancer.Availability and implementationA Python library that implements the proposed method is available at https://github.com/Mengyu8042/ot-grn.

The objective of this paper is to develop a duality between a novel entropy martingale optimal transport (EMOT) problem and an associated optimisation problem. In EMOT, we follow the approach taken in the entropy optimal transport (EOT) problem developed in Liero et al. (Invent. Math. 211:969–1117, 2018), but we add the constraint, typical of martingale optimal transport (MOT) theory, that the infimum of the cost functional is taken over martingale probability measures. In the associated problem, the objective functional, related via Fenchel conjugacy to the entropic term in EMOT, is no longer linear as in (martingale) optimal transport. This leads to a novel optimisation problem which also has a clear financial interpretation as a nonlinear subhedging problem. Our theory allows us to establish a nonlinear robust pricing–hedging duality which also covers a wide range of known robust results. We also focus on Wasserstein-induced penalisations and study how the duality is affected by variations in the penalty terms, with a special focus on the convergence of EMOT to the extreme case of MOT.

Treatment effect estimation helps answer questions, such as whether a specific treatment affects the outcome of interest. One fundamental issue in this research is to alleviate the treatment assignment bias among those treated units and controlled units. Classical causal inference methods resort to the propensity score estimation, which unfortunately tends to be misspecified when only limited overlapping exists between the treated and the controlled units. Moreover, existing supervised methods mainly consider the treatment assignment information underlying the factual space, and thus, their performance of counterfactual inference may be degraded due to overfitting of the factual results. To alleviate those issues, we build on the optimal transport theory and propose a novel causal optimal transport (CausalOT) model to estimate an individual treatment effect (ITE). With the proposed propensity measure, CausalOT can infer the counterfactual outcome by solving a novel regularized optimal transport problem, which allows the utilization of global information on observational covariates to alleviate the issue of limited overlapping. In addition, a novel counterfactual loss is designed for CausalOT to align the factual outcome distribution with the counterfactual outcome distribution. Most importantly, we prove the theoretical generalization bound for the counterfactual error of CausalOT. Empirical studies on benchmark datasets confirm that the proposed CausalOT outperforms state-of-the-art causal inference methods.

Maximum flow problem on dynamic generative network flows with time-varying bounds

The knowledge of the anatomical shape of both gross and microscopic structures is the key to understanding the effects of disease processes on cellular structure. Geometric morphometric methods, such as Procrustes superimposition, and Spherical Harmonics (SPHARM), have been used to capture the biological shape variation and group differences in morphology. Previous SPHARM-MAT techniques use the CALD algorithm to parameterize the mesh surface. It starts from initial mapping and performs local and global smoothing methods alternately to control the area and length distortions simultaneously. However, this parameterization may not be sufficient in complex morphological cases. To bridge this gap, we propose SPHARM-OT, an enhanced SPHARM surface modeling method using optimal transport (OT) for spherical parameterization. First, the genus 0 3D objects are conformally mapped onto a sphere. Then the optimal transport theory via spherical power diagram is introduced to minimize the area distortion. This new algorithm can effectively reduce the area distortion and lead to a better reconstruction result. We demonstrate the effectiveness of the method by applying it to the human sphenoidal paranasal sinuses.

To solve the poor real-time performance of the existing fault diagnosis algorithms on transmission system rotating components, this paper proposes a novel high-dimensional OT-Caps (Optimal Transport–Capsule Network) model. Based on the traditional capsule network algorithm, an auxiliary loss is introduced during the offline training process to improve the network architecture. Simultaneously, an optimal transport theory and a generative adversarial network are introduced into the auxiliary loss, which accurately depicts the error distribution of the fault characteristic. The proposed model solves the low real-time performance of the capsule network algorithm due to complex architecture, long calculation time, and oversized hardware resource consumption. Meanwhile, it ensures the high precision, early prediction, and transfer aptitude of fault diagnosis. Finally, the model’s effectiveness is verified by the public data sets and the actual faults data of the transmission system, which provide technical support for the application.

This paper re-visits Shayevitz & Feder's recent 'Posterior Matching Scheme', an explicit, dynamical system encoder for communication with feedback that treats the message as a point on the [0, 1] line and achieves capacity on memo-ryless channels. It has two key properties that ensure that it maximizes mutual information at each step: (a) the encoder sequentially hands the decoder what is missing; and (b) the next input has the desired statistics. Motivated by brain-machine interface applications and multi-antenna communications, we consider developing dynamical system feedback encoders for scenarios when the message point lies in higher dimensions. We develop a necessary and sufficient condition the Jacobian equation for any dynamical system encoder that maximizes mutual information. In general, there are many solutions to this equation. We connect this to the Monge-Kantorovich Optimal Transportation Problem, which provides a framework to identify a unique solution suiting a specific purpose. We provide two examplar capacity-achieving solutions for different purposes for the multi-antenna Gaussian channel with feedback. This insight further elucidates an interesting relationship between interactive decision theory problems and the theory of optimal transportation.