Optimal Transport Theory Research Articles

Combining Models of Root-Zone Hydrology and Geoelectrical Measurements: Recent Advances and Future Prospects

Recent advances in measuring and modeling root water uptake along with refined electrical petrophysical models may help fill the existing gap in hydrological root model parametrization. In this paper, we discuss the choices to be made to combine root-zone hydrology and geoelectrical data with the aim of characterizing the active root zone. For each model and observation type we discuss sources of uncertainty and how they are commonly addressed in a stochastic inversion framework. We point out different degrees of integration in the existing hydrogeophysical approaches to parametrize models of root-zone hydrology. This paper aims at giving emphasis to stochastic approaches, in particular to Data Assimilation (DA) schemes, that are generally identified as the best way to combine geoelectrical data with Root Water Uptake (RWU) models. In addition, the study points out a more suitable objective function taken from the optimal transport theory that better captures complex geometry of root systems. Another pathway for improvement of geoelectrical data integration into RWU models using DA relies on the use of stem based methods as a leverage to introduce more extensive root knowledge into RWU macroscopic hydrological models.

Independent Nonlinear Component Analysis

The idea of summarizing the information contained in a large number of variables by a small number of “factors” or “principal components” has been broadly adopted in statistics. This article introduces a generalization of the widely used principal component analysis (PCA) to nonlinear settings, thus providing a new tool for dimension reduction and exploratory data analysis or representation. The distinguishing features of the method include (i) the ability to always deliver truly independent (instead of merely uncorrelated) factors; (ii) the use of optimal transport theory and Brenier maps to obtain a robust and efficient computational algorithm; (iii) the use of a new multivariate additive entropy decomposition to determine the most informative principal nonlinear components, and (iv) formally nesting PCA as a special case for linear Gaussian factor models. We illustrate the method’s effectiveness in an application to excess bond returns prediction from a large number of macro factors. Supplementary materials for this article are available online.

Unsupervised CT Metal Artifact Learning Using Attention-Guided β-CycleGAN.

Metal artifact reduction (MAR) is one of the most important research topics in computed tomography (CT). With the advance of deep learning approaches for image reconstruction, various deep learning methods have been suggested for metal artifact reduction, among which supervised learning methods are most popular. However, matched metal-artifact-free and metal artifact corrupted image pairs are difficult to obtain in real CT acquisition. Recently, a promising unsupervised learning for MAR was proposed using feature disentanglement, but the resulting network architecture is so complicated that it is difficult to handle large size clinical images. To address this, here we propose a simple and effective unsupervised learning method for MAR. The proposed method is based on a novel β -cycleGAN architecture derived from the optimal transport theory for appropriate feature space disentanglement. Moreover, by adding the convolutional block attention module (CBAM) layers in the generator, we show that the metal artifacts can be more focused so that it can be effectively removed. Experimental results confirm that we can achieve improved metal artifact reduction that preserves the detailed texture of the original image.

Three-dimensional virtual histology of the human hippocampus based on phase-contrast computed tomography

We have studied the three-dimensional (3D) cytoarchitecture of the human hippocampus in neuropathologically healthy and Alzheimer's disease (AD) individuals, based on phase-contrast X-ray computed tomography of postmortem human tissue punch biopsies. In view of recent findings suggesting a nuclear origin of AD, we target in particular the nuclear structure of the dentate gyrus (DG) granule cells. Tissue samples of 20 individuals were scanned and evaluated using a highly automated approach of measurement and analysis, combining multiscale recordings, optimized phase retrieval, segmentation by machine learning, representation of structural properties in a feature space, and classification based on the theory of optimal transport. Accordingly, we find that the prototypical transformation between a structure representing healthy granule cells and the pathological state involves a decrease in the volume of granule cell nuclei, as well as an increase in the electron density and its spatial heterogeneity. The latter can be explained by a higher ratio of heterochromatin to euchromatin. Similarly, many other structural properties can be derived from the data, reflecting both the natural polydispersity of the hippocampal cytoarchitecture between different individuals in the physiological context and the structural effects associated with AD pathology.

Surrogate parametric metamodel based on Optimal Transport

The description of a physical problem through a model necessarily involves the introduction of parameters. Hence, one wishes to have a solution of the problem that is a function of all these parameters: a parametric solution. However, the construction of such parametric solutions exhibiting localization in space is only ensured by costly and time-consuming tests, which can be both numerical or experimental. Numerical methodologies used classically imply enormous computational efforts for exploring the design space. Therefore, parametric solutions obtained using advanced nonlinear regressions are an essential tool to address this challenge. However, classical regression techniques, even the most advanced ones, can lead to non physical interpolation in some fields such as fluid dynamics, where the solution localizes in different regions depending on the problem parameters choice. In this context, Optimal Transport (OT) offers a mathematical approach to measure distances and interpolate between general objects in a, sometimes, more physical way than the classical interpolation approach. Thus, OT has become fundamental in some fields such as statistics or computer vision, and it is being increasingly used in fields such as computational mechanics. However, the OT problem is usually computationally costly to solve and not adapted to be accessed in an online manner. Therefore, the aim of this paper is combining advanced nonlinear regressions with Optimal Transport in order to implement a parametric real-time model based on OT. To this purpose, a parametric model is built offline relying on Model Order Reduction and OT, leading to a real-time interpolation tool following Optimal Transport theory. Such a tool is of major interest in design processes, but also within the digital twin rationale.

Blaschke–Santaló inequality for many functions and geodesic barycenters of measures

Motivated by the geodesic barycenter problem from optimal transportation theory, we prove a natural generalization of the Blaschke–Santaló inequality and the affine isoperimetric inequalities for many sets and many functions. We derive from it an entropy bound for the total Kantorovich cost appearing in the barycenter problem. We also establish a “pointwise Prékopa–Leindler inequality” and show a monotonicity property of the multimarginal Blaschke–Santaó functional.

Partial optimal transport for a constant-volume Lagrangian mesh with free boundaries

This article introduces a representation of dynamic meshes, adapted to some numerical simulations that require controlling the volume of objects with free boundaries, such as incompressible fluid simulation, some astrophysical simulations at cosmological scale, and shape/topology optimization. The algorithm decomposes the simulated object into a set of convex cells called a Laguerre diagram, parameterized by the position of N points in 3D and N additional parameters that control the volumes of the cells. These parameters are found as the (unique) solution of a convex optimization problem – semi-discrete Monge-Ampère equation – stemming from optimal transport theory. In this article, this setting is extended to objects with free boundaries and arbitrary topology, evolving in a domain of arbitrary shape, by solving a partial optimal transport problem. The resulting Lagrangian scheme makes it possible to accurately control the volume of the object, while precisely computing the intersections with the domain boundary, the interactions, the collisions, and the changes of topology.

Geometrical aspects of entropy production in stochastic thermodynamics based on Wasserstein distance

We study a relationship between optimal transport theory and stochastic thermodynamics for the Fokker-Planck equation. We show that the lower bound on the entropy production is the action measured by the path length of the $L^2$-Wasserstein distance. Because the $L^2$-Wasserstein distance is a geometric measure of optimal transport theory, our result implies a geometric interpretation of the entropy production. Based on this interpretation, we obtain a thermodynamic trade-off relation between transition time and the entropy production. This thermodynamic trade-off relation is regarded as a thermodynamic speed limit which gives a tighter bound of the entropy production. We also discuss stochastic thermodynamics for the subsystem and derive a lower bound on the partial entropy production as a generalization of the second law of information thermodynamics. Our formalism also provides a geometric picture of the optimal protocol to minimize the entropy production. We illustrate these results by the optimal stochastic heat engine and show a geometrical bound of the efficiency.

Probabilistic measures of edge criticality in graphs: a study in water distribution networks

The issue of vulnerability and robustness in networks have been addressed by several methods. The goal is to identify which are the critical components (i.e., nodes/edges) whose failure impairs the functioning of the network and how much this impacts the ensuing increase in vulnerability. In this paper we consider the drop in the network robustness as measured by the increase in vulnerability of the perturbed network and compare it with the original one. Traditional robustness metrics are based on centrality measures, the loss of efficiency and spectral analysis. The approach proposed in this paper sees the graph as a set of probability distributions and computes, specifically the probability distribution of its node to node distances and computes an index of vulnerability through the distance between the node-to-node distributions associated to original network and the one obtained by the removal of nodes and edges. Two such distances are proposed for this analysis: Jensen–Shannon and Wasserstein, based respectively on information theory and optimal transport theory, which are shown to offer a different characterization of vulnerability. Extensive computational results, including two real-world water distribution networks, are reported comparing the new approach to the traditional metrics. This modelling and algorithmic framework can also support the analysis of other networked infrastructures among which power grids, gas distribution and transit networks.

An Optimal Transport Based Transferable System for Detection of Erroneous Somato-Sensory Feedback from Neural Signals.

This study is aimed at the detection of single-trial feedback, perceived as erroneous by the user, using a transferable classification system while conducting a motor imagery brain–computer interfacing (BCI) task. The feedback received by the users are relayed from a functional electrical stimulation (FES) device and hence are somato-sensory in nature. The BCI system designed for this study activates an electrical stimulator placed on the left hand, right hand, left foot, and right foot of the user. Trials containing erroneous feedback can be detected from the neural signals in form of the error related potential (ErrP). The inclusion of neuro-feedback during the experiments indicated the possibility that ErrP signals can be evoked when the participant perceives an error from the feedback. Hence, to detect such feedback using ErrP, a transferable (offline) decoder based on optimal transport theory is introduced herein. The offline system detects single-trial erroneous trials from the feedback period of an online neuro-feedback BCI system. The results of the FES-based feedback BCI system were compared to a similar visual-based (VIS) feedback system. Using our framework, the error detector systems for both the FES and VIS feedback paradigms achieved an F1-score of 92.66% and 83.10%, respectively, and are significantly superior to a comparative system where an optimal transport was not used. It is expected that this form of transferable and automated error detection system compounded with a motor imagery system will augment the performance of a BCI and provide a better BCI-based neuro-rehabilitation protocol that has an error control mechanism embedded into it.

Designing optimal networks for multicommodity transport problem

Designing and optimizing different flows in networks is a relevant problem in many contexts. While a number of methods have been proposed in the physics and optimal transport literature for the one-commodity case, we lack similar results for the multicommodity scenario. In this paper we present a model based on optimal transport theory for finding optimal multicommodity flow configurations on networks. This model introduces a dynamics that regulates the edge conductivities to achieve, at infinite times, a minimum of a Lyapunov functional given by the sum of a convex transport cost and a concave infrastructure cost. We show that the long-time asymptotics of this dynamics are the solutions of a standard constrained optimization problem that generalizes the one-commodity framework. Our results provide insights into the nature and properties of optimal network topologies. In particular, they show that loops can arise as a consequence of distinguishing different flow types, complementing previous results where loops, in the one-commodity case, were obtained as a consequence of imposing dynamical rules on the sources and sinks or when enforcing robustness to damage. Finally, we provide an efficient implementation of our model which converges faster than standard optimization methods based on gradient descent.

Modeling COVID-19 with mean field evolutionary dynamics: Social distancing and seasonality

The coronavirus pandemic has been declared a world health emergency by the World Health Organization, which has raised the importance of an accurate epidemiological model to predict the evolution of COVID-19. In this paper, we propose mean field evolutionary dynamics (MFEDs), inspired by optimal transport theory and mean field games on graphs, to model the evolution of COVID-19. In the MFEDs, we derive the payoff functions for different individual states from the commonly used replicator dynamics (RDs) and employ them to govern the evolution of epidemics. We also compare epidemic modeling based on MFEDs with that based on RDs through numerical experiments. Moreover, we show the efficiency of the proposed MFED-based model by fitting it to the COVID-19 statistics of Wuhan, China. Finally, we analyze the effects of one-time social distancing as well as the seasonality of COVID-19 through the post-pandemic period.

Sliced Wasserstein based Canonical Correlation Analysis for Cross-Domain Recommendation

• A cross-domain recommendation model based on Sliced Wasserstein autoencoder is proposed. • An improved cross-domain transformation loss of orthogonal transformation is proposed. • Rigorous experiments were carried out to demonstrate the efficiency of the model. To solve the problem of data sparsity and cold start, the cross-domain recommendation is a promising research direction in the recommender system . The goal of cross-domain recommendation is to transfer learned knowledge from the source domain to the target domain by different means to improve the performance of the recommendation. But most approaches face the distribution misalignment . In this paper, we propose a joint learning cross-domain recommendation model that can extract domain-specific and common features simultaneously, and only use the implicit feedback data of users without additional auxiliary information. To the best of our knowledge, it is the first attempt to combine the sliced Wasserstein distance and canonical correlation analysis for the cross-domain recommendation scenario. Our one intuition is to reduce the reconstruction error caused by the variational inference based autoencoder model by the optimal transportation theory. Another attempt is to improve the correlation between domains by combining the idea of the canonical correlation analysis. With rigorous experiments, we empirically demonstrated that our model can achieve better performance compared with the state-of-the-art methods.

Cooperative communication as belief transport.

Recent research formalizes cooperative communication as belief transport using the mathematical theory of optimal transport. This formalization allows rigorous a priori analysis of the statistical and ecological properties of models of cooperative communication, unification of prior models and analysis of their differences, and promising directions for future research.

Density-aware decentralised multi-agent exploration with energy constraint based on optimal transport theory

This paper addresses a density-aware multi-agent exploration problem based on an Optimal Transport (OT) theory while considering energy constraints of a multi-agent system. The density-aware exploration means how a team of agents (robots) cover a given domain, reflecting a priority of areas of interest represented by a density distribution, rather than simply following a preset of uniform patterns. To achieve the density-aware multi-agent exploration, the optimal transport theory that quantifies a distance between two density distributions is employed as a tool, which also serves as a means of similarity measure. Energy constraints for a multi-agent system are then incorporated into the OT-based density-aware multi-agent exploration scheme. The proposed method is developed targeting a decentralised control to cope with more realistic scenarios such as communication range limits between agents. To measure the exploration efficiency, the upper bound of the similarity measure is proposed, which is computationally tractable. The developed multi-agent exploration scheme is applicable to a time-varying distribution as well, where a spatio-temporal evolution of the given reference distribution is desired. To validate the proposed method, multiple simulation results are provided.

Optimal transport of vector measures

We develop and study a theory of optimal transport for vector measures. We resolve in the negative a conjecture of Klartag, that given a vector measure on Euclidean space with total mass zero, the mass of any transport set is again zero. We provide a counterexample to the conjecture. We generalise the Kantorovich–Rubinstein duality to the vector measures setting. Employing the generalisation, we answer the conjecture in the affirmative provided there exists an optimal transport with absolutely continuous marginals of its total variation.

Schrödinger dynamics and optimal transport of measures

In this paper, we recover a class of displacement interpolations of probability measures, in the sense of the Optimal Transport theory, by means of semiclassical measures associated with solutions of Schrödinger equation defined on the flat torus. Moreover, we prove the completing viewpoint by proving that a family of displacement interpolations can always be viewed as a path of time-dependent semiclassical measures.

Balanced distortion and perception in single-image super-resolution based on optimal transport in wavelet domain

Single image super-resolution (SISR) is a classic ill-posed problem in computer vision. In recent years, deep-learning-based (DL-based) models have achieved promising results with the SISR problem. However, most existing methods suffer from an intrinsic trade-off between distortion and perceptual quality. To satisfy the requirements in different real-world situations, the balance of distortion and visual quality for image super-resolution is a critical issue. In DL-based models, the uses of hybrid loss (i.e., the combination of the distortion loss and the perceptual loss) and network interpolation are two common approaches to balancing the distortion and perceptual quality of super-resolved images. However, these two kinds of methods lack flexibility and hold strict constraints on network architectures. In this paper, we propose an image-fusion interpolation method for image super-resolution, which can balance the distortion and visual quality of super-resolved images, based on the optimal transport theory in the wavelet domain. The advantage of our proposed method is that it can be applied to any pretrained DL-based model, without any requirement from the network architecture and parameters. In addition, our proposed method is parameter-free and can run fast without using a GPU. Compared with existing state-of-the-art SISR methods, experiment results show that our proposed method can achieve a better balance between the distortion and visual quality in super-resolved images.

3D brain tumor segmentation using a two-stage optimal mass transport algorithm

Optimal mass transport (OMT) theory, the goal of which is to move any irregular 3D object (i.e., the brain) without causing significant distortion, is used to preprocess brain tumor datasets for the first time in this paper. The first stage of a two-stage OMT (TSOMT) procedure transforms the brain into a unit solid ball. The second stage transforms the unit ball into a cube, as it is easier to apply a 3D convolutional neural network to rectangular coordinates. Small variations in the local mass-measure stretch ratio among all the brain tumor datasets confirm the robustness of the transform. Additionally, the distortion is kept at a minimum with a reasonable transport cost. The original 240 times 240 times 155 times 4 dataset is thus reduced to a cube of 128 times 128 times 128 times 4, which is a 76.6% reduction in the total number of voxels, without losing much detail. Three typical U-Nets are trained separately to predict the whole tumor (WT), tumor core (TC), and enhanced tumor (ET) from the cube. An impressive training accuracy of 0.9822 in the WT cube is achieved at 400 epochs. An inverse TSOMT method is applied to the predicted cube to obtain the brain results. The conversion loss from the TSOMT method to the inverse TSOMT method is found to be less than one percent. For training, good Dice scores (0.9781 for the WT, 0.9637 for the TC, and 0.9305 for the ET) can be obtained. Significant improvements in brain tumor detection and the segmentation accuracy are achieved. For testing, postprocessing (rotation) is added to the TSOMT, U-Net prediction, and inverse TSOMT methods for an accuracy improvement of one to two percent. It takes 200 seconds to complete the whole segmentation process on each new brain tumor dataset.

Modeling Epidemic Spread among a Commuting Population Using Transport Schemes

Understanding the dynamics of the spread of COVID-19 between connected communities is fundamental in planning appropriate mitigation measures. To that end, we propose and analyze a novel metapopulation network model, particularly suitable for modeling commuter traffic patterns, that takes into account the connectivity between a heterogeneous set of communities, each with its own infection dynamics. In the novel metapopulation model that we propose here, transport schemes developed in optimal transport theory provide an efficient and easily implementable way of describing the temporary population redistribution due to traffic, such as the daily commuter traffic between work and residence. Locally, infection dynamics in individual communities are described in terms of a susceptible-exposed-infected-recovered (SEIR) compartment model, modified to account for the specific features of COVID-19, most notably its spread by asymptomatic and presymptomatic infected individuals. The mathematical foundation of our metapopulation network model is akin to a transport scheme between two population distributions, namely the residential distribution and the workplace distribution, whose interface can be inferred from commuter mobility data made available by the US Census Bureau. We use the proposed metapopulation model to test the dynamics of the spread of COVID-19 on two networks, a smaller one comprising 7 counties in the Greater Cleveland area in Ohio, and a larger one consisting of 74 counties in the Pittsburgh–Cleveland–Detroit corridor following the Lake Erie’s American coastline. The model simulations indicate that densely populated regions effectively act as amplifiers of the infection for the surrounding, less densely populated areas, in agreement with the pattern of infections observed in the course of the COVID-19 pandemic. Computed examples show that the model can be used also to test different mitigation strategies, including one based on state-level travel restrictions, another on county level triggered social distancing, as well as a combination of the two.