Pediatric Sarcoma Data Forms a Unique Cluster Measured via the Earth Mover’s Distance

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 work, we develop a new framework for dynamic network flow problems based on optimal transport theory. We show that the dynamic multicommodity minimum-cost network flow problem can be formulated as a multimarginal optimal transport problem, where the cost function and the constraints on the marginals are associated with a graph structure. By exploiting these structures and building on recent advances in optimal transport theory, we develop an efficient method for such entropy-regularized optimal transport problems. In particular, the graph structure is utilized to efficiently compute the projections needed in the corresponding Sinkhorn iterations, and we arrive at a scheme that is both highly computationally efficient and easy to implement. To illustrate the performance of our algorithm, we compare it with a state-of-the-art linear programming (LP) solver. We achieve good approximations to the solution at least one order of magnitude faster than the LP solver. Finally, we showcase the methodology on a traffic routing problem with a large number of commodities. Funding: This work was supported by KTH Digital Futures, Knut och Alice Wallenbergs Stiftelse [Grants KAW 2018.0349, KAW 2021.0274, the Wallenberg AI, Autonomous Systems and Software Program (WASP) funded by the Knut and Alice Wallenberg Foundation], Vetenskapsrådet [Grant 2020-03454], and the National Science Foundation [Grants 1942523 and 2206576].

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.

Merging databases is a strategy of paramount interest especially in medical research. A common problem in this context comes from a variable which is not coded on the same scale in both databases we aim to merge. This paper considers the problem of finding a relevant way to recode the variable in order to merge these two databases. To address this issue, an algorithm, based on optimal transportation theory, is proposed. Optimal transportation theory gives us an application to map the measure associated with the variable in database A to the measure associated with the same variable in database B. To do so, a cost function has to be introduced and an allocation rule has to be defined. Such a function and such a rule is proposed involving the information contained in the covariates. In this paper, the method is compared to multiple imputation by chained equations and a statistical learning method and has demonstrated a better average accuracy in many situations. Applications on both simulated and real datasets show that the efficiency of the proposed merging algorithm depends on how the covariates are linked with the variable of interest.

In this paper, the effective use of flight-time constrained unmanned aerial vehicles (UAVs) as flying base stations that provide wireless service to ground users is investigated. In particular, a novel framework for optimizing the performance of such UAV-based wireless systems in terms of the average number of bits (data service) transmitted to users as well as the UAVs' hover duration (i.e. flight time) is proposed. In the considered model, UAVs hover over a given geographical area to serve ground users that are distributed within the area based on an arbitrary spatial distribution function. In this case, two practical scenarios are considered. In the first scenario, based on the maximum possible hover times of UAVs, the average data service delivered to the users under a fair resource allocation scheme is maximized by finding the optimal cell partitions associated to the UAVs. Using the powerful mathematical framework of optimal transport theory, this cell partitioning problem is proved to be equivalent to a convex optimization problem. Subsequently, a gradient-based algorithm is proposed for optimally partitioning the geographical area based on the users' distribution, hover times, and locations of the UAVs. In the second scenario, given the load requirements of ground users, the minimum average hover time that the UAVs need for completely servicing their ground users is derived. To this end, first, an optimal bandwidth allocation scheme for serving the users is proposed. Then, given this optimal bandwidth allocation, optimal cell partitions associated with the UAVs are derived by exploiting the optimal transport theory. Simulation results show that our proposed cell partitioning approach leads to a significantly higher fairness among the users compared with the classical weighted Voronoi diagram. Furthermore, the results demonstrate that the average hover time of the UAVs can be reduced by 64% by adopting the proposed optimal bandwidth allocation scheme as well as the optimal cell partitioning approach. In addition, our results reveal an inherent tradeoff between the hover time of UAVs and bandwidth efficiency while serving the ground users.

Markov State Models (MSMs) describe the rates and routes in conformational dynamics of biomolecules. Computational estimation of MSMs can be expensive becausemolecular simulations are slow to �nd and sample the rare transient events. We describe here an ecient approximate way to determine MSM rate matrices by combining Maximum Caliber (maximizing path entropies) with Optimal Transport Theory (minimizing some path cost function, as when routing trucks on transportationnetworks) to patch together transient dynamical information from multiple nonequilibriumsimulations. We give toy examples.

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.

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.

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.

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.

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.

We analyze the distinguishability of two sources in a Neyman-Pearson setup when an attacker is allowed to modify the output of one of the two sources subject to an additive distortion constraint. By casting the problem in a game-theoretic framework and by exploiting the parallelism between the attacker's goal and optimal transport theory, we introduce the concept of security margin defined as the maximum average per-sample distortion introduced by the attacker for which the two sources can be distinguished ensuring arbitrarily small, yet positive, error exponents for type I and type II error probabilities. Several versions of the problem are considered according to the available knowledge about the sources. We compute the security margin for some classes of sources and derive general bounds assuming that the distortion is measured in terms of the mean square error between the original and the attacked sequence. The analysis of the game and the study of the distinguishability of the sources are extended to the case in which the distortion constraint is defined in terms of the maximum distance.

Polynomial chaos (PC) expansions are used for the propagation of uncertainty through dynamical systems as an alternative to Monte Carlo methods. Model parameters in a given dynamical system are assumed to have known expansions, which correspond to simple standard distributions, and one is usually interested in the polynomial expansion of the system solution. We are concerned with the problem of estimating the PC expansion of a parameter vector when only realizations from its distribution are given. To this end we apply ideas from optimal transportation theory and network optimization.

Wasserstein distances in optimal transport provides a mathematical tool to measure distances between functions or more general objects. By Wasserstein distances, we define a distance on the moduli space of a class of statistical manifolds. We construct a Riemannian metric of this space and verify that the defined distance can be regarded as the induced distance of the metric.

SummaryWe develop an optimal transportation mesh‐free particle method for advection‐diffusion in which the concentration or density of the diffusive species is approximated by Dirac measures. We resort to an incremental variational principle for purposes of time discretization of the diffusive step. This principle characterizes the evolution of the density as a competition between the Wasserstein distance between two consecutive densities and entropy. Exploiting the structure of the Euler–Lagrange equations, we approximate the density as a collection of Diracs. The interpolation of the incremental transport map is effected through mesh‐free max‐ent interpolation. Remarkably, the resulting update is geometrically exact with respect to advection and volume. We present three‐dimensional examples of application that illustrate the scope and robustness of the method. Copyright © 2017 John Wiley & Sons, Ltd.