Optimal Transport Plan Research Articles

Alignment and comparison of directed networks via transition couplings of random walks

Abstract We describe and study a transport-based procedure called network optimal transition coupling (NetOTC) for the comparison and alignment of two networks. The networks of interest may be directed or undirected, weighted or unweighted, and may have distinct vertex sets of different sizes. Given two networks and a cost function relating their vertices, NetOTC finds a transition coupling of their associated random walks having minimum expected cost. The minimizing cost quantifies the difference between the networks, while the optimal transport plan itself provides alignments of both the vertices and the edges of the two networks. Coupling of the full random walks, rather than their marginal distributions, ensures that NetOTC captures local and global information about the networks and preserves edges. NetOTC has no free parameters and does not rely on randomization. We investigate a number of theoretical properties of NetOTC and present experiments establishing its empirical performance.

Maximizers of nonlocal interactions of Wasserstein Type

We characterize the maximizers of a functional that involves the minimization of the Wasserstein distance between sets of equal volume. We prove that balls are the only maximizers by combining a symmetrization-by-reflection technique with the uniqueness of optimal transport plans. Further, in one dimension, we provide a sharp quantitative refinement of this maximality result.

OTMorph: Unsupervised Multi-domain Abdominal Medical Image Registration Using Neural Optimal Transport.

Deformable image registration is one of the essential processes in analyzing medical images. In particular, when diagnosing abdominal diseases such as hepatic cancer and lymphoma, multi-domain images scanned from different modalities or different imaging protocols are often used. However, they are not aligned due to scanning times, patient breathing, movement, etc. Although recent learning-based approaches can provide deformations in real-time with high performance, multi-domain abdominal image registration using deep learning is still challenging since the images in different domains have different characteristics such as image contrast and intensity ranges. To address this, this paper proposes a novel unsupervised multi-domain image registration framework using neural optimal transport, dubbed OTMorph. When moving and fixed volumes are given as input, a transport module of our proposed model learns the optimal transport plan to map data distributions from the moving to the fixed volumes and estimates a domain-transported volume. Subsequently, a registration module taking the transported volume can effectively estimate the deformation field, leading to deformation performance improvement. Experimental results on multi-domain image registration using multi-modality and multi-parametric abdominal medical images demonstrate that the proposed method provides superior deformable registration via the domain-transported image that alleviates the domain gap between the input images. Also, we attain the improvement even on out-of-distribution data, which indicates the superior generalizability of our model for the registration of various medical images. Our source code is available at https://github.com/boahK/OTMorph.

Bilevel Optimization of the Kantorovich Problem and Its Quadratic Regularization

This paper is concerned with an optimization problem which is governed by the Kantorovich problem of optimal transport. More precisely, we consider a bilevel optimization problem with the underlying problem being the Kantorovich problem. This task can be reformulated as a mathematical problem with complementarity constraints in the space of regular Borel measures. Because of the non-smoothness that is induced by the complementarity constraints, problems of this type are often regularized, e.g., by an entropic regularization. However, in this paper we apply a quadratic regularization to the Kantorovich problem. By doing so, we are able to drastically reduce its dimension while preserving the sparsity structure of the optimal transportation plan as much as possible. As the title indicates, this is the second part in a series of three papers. While the existence of optimal solutions to both the bilevel Kantorovich problem and its regularized counterpart were shown in the first part, this paper deals with the (weak-∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$*$$\\end{document}) convergence of solutions to the regularized bilevel problem to solutions of the original bilevel Kantorovich problem for vanishing regularization parameters.

A distributionally robust optimization approach for the potassium fertilizer product transportation considering transshipment through crossdocks

Potassium fertilizer is an essential input for agricultural productivity, and plays a critical role in various plant processes, influencing water uptake, enzyme activation, and photosynthesis. The efficient delivery of potassium fertilizer products to the end-users, typically farmers and agricultural enterprises, is of utmost importance. Due to the long traveling distance between the supplier and customers, decision makers of supplier normally set up multiple crossdocks to aid in transition and storage of potassium fertilizer products. In this situation, making an optimal transportation plan as well as the inventory level of crossdocks will directly enhance the efficiency and effectiveness of long-distance transportation. In this study, we model this problem as a dynamic transportation problem with transshipment considering the uncertain time-varying customers’ demand. To characterize the demand information, we first construct an ambiguity set of customers’ demand based in limited historical data and then develop a distributionally robust optimization-based (DRO) framework to optimize the transportation plan and related inventory level of crossdocks simultaneously. We also propose a general approach to overcome the computational challenges of DRO by transforming the original DRO into a second-order cone programming based on duality theory. Additionally, we introduce linear decision rules to adjust the optimization strategy based on the new observed demand, thus lead the model to handle the dynamic information flow in real time. In case study, all involved data are collected or derived from a real potassium fertilizer company located at Western China. The results show our model reduces the cost by 18.07% compared to stochastic model (sample average approximation), indicating a significant effectiveness of our model on the improvement of delivery efficiency and cost saving in real dynamic logistic systems. Also, we conclude the managerial insight that decision-makers should develop a comprehensive strategy, including improving communication to ensure order status updates, planning rationally to evenly distribute orders, and proactively allocating resources to meet operational demands.

Optimized transportation scheduling for precast concrete components considering heterogeneous vehicle-size matching

Recently, prefabricated construction has been vigorously promoted, resulting in high demand for precast concrete (PC) components. The transportation scheduling optimization problem of PC components with various kinds from multiple projects arises. Unlike conventional cargo, PC components are characterized by shape heterogeneity, large volume, and strict delivery time limits. Based on three characteristics, a heterogeneous fixed fleet vehicle routing problem (HFFVRP) for PC components is introduced, where heterogeneous vehicles, allocation of PC components to size-matching vehicles, and hybrid time windows are considered. Then a two-stage solution strategy based on the improved ant colony optimization (ACO) and Dijkstra algorithm is designed to obtain optimal vehicle routes under minimum transportation costs. The results indicate that the improved ACO-Dijkstra algorithm outperforms in obtaining optimal transportation plans for heterogeneous vehicles compared with manual decision-making and other heuristic algorithms. Sensitivity analysis denotes that utilizing heterogeneous vehicles contributes to reductions in transportation costs, and vehicle configuration should be adjusted along with demand scales. The proposed model and algorithm extend the theoretical basis of construction industrial applications.

On Stability Estimate of Optimal Transport Maps Using m-th Polynomial Convexity

In 1781, Gaspard Monge first proposed the practical problem of relocating building materials while minimizing workers’ effort. Mathematically, the problem can be reiterated as finding a mapping T0 that transforms a random variable (X) following probability measure (μ) into a random variable (Y) following probability measure (ν), with minimal cost. Afterward, it has been widely studied and applied in statistics, machine learning, and economics, which concern the study of “distance” between usually a pair of probability distributions. The focus of this paper is centered on investigating and generalizing stability estimates for optimal transport plans, particularly through the lens of strong polynomial convexity. Building on previous research using plug-in estimators to strengthen the convergence rate of discrete or semi-discrete estimators for optimal transport plans, this paper introduces a novel stability estimate leveraging L-Lipschitz continuity and a paradigmatic methodology based on polynomial convexity, the understanding of which remains limited.

Sufficiency of -cyclical monotonicity in a class of multi-marginal optimal transport problems

Abstract $c$ -cyclical monotonicity is the most important optimality condition for an optimal transport plan. While the proof of necessity is relatively easy, the proof of sufficiency is often more difficult or even elusive. We present here a new approach, and we show how known results are derived in this new framework and how this approach allows to prove sufficiency in situations previously not treatable.

Entropic transfer operators

We propose a new concept for the regularization and discretization of transfer and Koopman operators in dynamical systems. Our approach is based on the entropically regularized optimal transport between two probability measures. In particular, we use optimal transport plans in order to construct a finite-dimensional approximation of some transfer or Koopman operator which can be analyzed computationally. We prove that the spectrum of the discretized operator converges to the one of the regularized original operator, give a detailed analysis of the relation between the discretized and the original peripheral spectrum for a rotation map on the n-torus and provide code for three numerical experiments, including one based on the raw trajectory data of a small biomolecule from which its dominant conformations are recovered.

Unsupervised Multi-parametric MRI Registration Using Neural Optimal Transport.

Precise deformable image registration of multi-parametric MRI sequences is necessary for radiologists in order to identify abnormalities and diagnose diseases, such as prostate cancer and lymphoma. Despite recent advances in unsupervised learning-based registration, volumetric medical image registration that requires considering the variety of data distributions is still challenging. To address the problem of multi-parametric MRI sequence data registration, we propose an unsupervised domain-transported registration method, called OTMorph by employing neural optimal transport that learns an optimal transport plan to map different data distributions. We have designed a novel framework composed of a transport module and a registration module: the former transports data distribution from the moving source domain to the fixed target domain, and the latter takes the transported data and provides the deformed moving volume that is aligned with the fixed volume. Through end-to-end learning, our proposed method can effectively learn deformable registration for the volumes in different distributions. Experimental results with abdominal multi-parametric MRI sequence data show that our method has superior performance over around 67-85% in deforming the MRI volumes compared to the existing learning-based methods. Our method is generic in nature and can be used to register inter-/intra-modality images by mapping the different data distributions in network training.

Estimation of stationary optimal transport plans

Abstract We study optimal transport for stationary stochastic processes taking values in finite spaces. In order to reflect the stationarity of the underlying processes, we restrict attention to stationary couplings, also known as joinings. The resulting optimal joining problem captures differences in the long-run average behavior of the processes of interest. We introduce estimators of both optimal joinings and the optimal joining cost, and establish consistency of the estimators under mild conditions. Furthermore, under stronger mixing assumptions we establish finite-sample error rates for the estimated optimal joining cost that extend the best known results in the iid case. We also extend the consistency and rate analysis to an entropy-penalized version of the optimal joining problem. Finally, we validate our convergence results empirically as well as demonstrate the computational advantage of the entropic problem in a simulation experiment.

Convergence proof for the GenCol algorithm in the case of two-marginal optimal transport

The recently introduced Genetic Column Generation (GenCol) algorithm has been numerically observed to efficiently and accurately compute high-dimensional optimal transport (OT) plans for general multi-marginal problems, but theoretical results on the algorithm have hitherto been lacking. The algorithm solves the OT linear program on a dynamically updated low-dimensional submanifold consisting of sparse plans. The submanifold dimension exceeds the sparse support of optimal plans only by a fixed factor β \beta . Here we prove that for β ≥ 2 \beta \geq 2 and in the two-marginal case, GenCol always converges to an exact solution, for arbitrary costs and marginals. The proof relies on the concept of c-cyclical monotonicity. As an offshoot, GenCol rigorously reduces the data complexity of numerically solving two-marginal OT problems from O ( ℓ 2 ) O(\ell ^2) to O ( ℓ ) O(\ell ) without any loss in accuracy, where ℓ \ell is the number of discretization points for a single marginal. At the end of the paper we also present some insights into the convergence behavior in the multi-marginal case.

Probability-Polarized Optimal Transport for Unsupervised Domain Adaptation

Optimal transport (OT) is an important methodology to measure distribution discrepancy, which has achieved promising performance in artificial intelligence applications, e.g., unsupervised domain adaptation. However, from the view of transportation, there are still limitations: 1) the local discriminative structures for downstream tasks, e.g., cluster structure for classification, cannot be explicitly admitted by the learned OT plan; 2) the entropy regularization induces a dense OT plan with increasing uncertainty. To tackle these issues, we propose a novel Probability-Polarized OT (PPOT) framework, which can characterize the structure of OT plan explicitly. Specifically, the probability polarization mechanism is proposed to guide the optimization direction of OT plan, which generates a clear margin between similar and dissimilar transport pairs and reduces the uncertainty. Further, a dynamic mechanism for margin is developed by incorporating task-related information into the polarization, which directly captures the intra/inter class correspondence for knowledge transportation. A mathematical understanding for PPOT is provided from the view of gradient, which ensures interpretability. Extensive experiments on several datasets validate the effectiveness and empirical efficiency of PPOT.

A Novel Skip Orthogonal List for Dynamic Optimal Transport Problem

Optimal transport is a fundamental topic that has attracted a great amount of attention from the optimization community in the past decades. In this paper, we consider an interesting discrete dynamic optimal transport problem: can we efficiently update the optimal transport plan when the weights or the locations of the data points change? This problem is naturally motivated by several applications in machine learning. For example, we often need to compute the optimal transport cost between two different data sets; if some changes happen to a few data points, should we re-compute the high complexity cost function or update the cost by some efficient dynamic data structure? We are aware that several dynamic maximum flow algorithms have been proposed before, however, the research on dynamic minimum cost flow problem is still quite limited, to the best of our knowledge. We propose a novel 2D Skip Orthogonal List together with some dynamic tree techniques. Although our algorithm is based on the conventional simplex method, it can efficiently find the variable to pivot within expected O(1) time, and complete each pivoting operation within expected O(|V|) time where V is the set of all supply and demand nodes. Since dynamic modifications typically do not introduce significant changes, our algorithm requires only a few simplex iterations in practice. So our algorithm is more efficient than re-computing the optimal transport cost that needs at least one traversal over all |E|=O(|V|^2) variables, where |E| denotes the number of edges in the network. Our experiments demonstrate that our algorithm significantly outperforms existing algorithms in the dynamic scenarios.

Convergence rate of entropy-regularized multi-marginal optimal transport costs

Abstract We investigate the convergence rate of multi-marginal optimal transport costs that are regularized with the Boltzmann–Shannon entropy, as the noise parameter $\varepsilon $ tends to $0$ . We establish lower and upper bounds on the difference with the unregularized cost of the form $C\varepsilon \log (1/\varepsilon )+O(\varepsilon )$ for some explicit dimensional constants C depending on the marginals and on the ground cost, but not on the optimal transport plans themselves. Upper bounds are obtained for Lipschitz costs or locally semiconcave costs for a finer estimate, and lower bounds for $\mathscr {C}^2$ costs satisfying some signature condition on the mixed second derivatives that may include degenerate costs, thus generalizing results previously in the two marginals case and for nondegenerate costs. We obtain in particular matching bounds in some typical situations where the optimal plan is deterministic.

Optimal transport-based machine learning to match specific patterns: application to the detection of molecular regulation patterns in omics data

Abstract We present several algorithms designed to learn a pattern of correspondence between 2 data sets in situations where it is desirable to match elements that exhibit a relationship belonging to a known parametric model. In the motivating case study, the challenge is to better understand micro-RNA regulation in the striatum of Huntington’s disease model mice. The algorithms unfold in 2 stages. First, an optimal transport plan P and an optimal affine transformation are learned, using the Sinkhorn–Knopp algorithm and a mini-batch gradient descent. Second, P is exploited to derive either several co-clusters or several sets of matched elements. A simulation study illustrates how the algorithms work and perform. The real data application further illustrates their applicability and interest.

A dragonfly algorithm for solving the Fixed Charge Transportation Problem FCTP

The primary focus of this article is dedicated to a thorough investigation of the Fixed Load Transportation Problem (FCTP) and the proposition of an exceedingly efficient resolution method, with a specific emphasis on the achievement of optimal transportation plans within practical time constraints. The FCTP, recognized for its intricate nature, falls into the NP-complete category, notorious for its exponential growth in solution time as the problem's size escalates. Within the realm of combinatorial optimization, metaheuristic techniques like the Dragonfly algorithm and genetic algorithms have garnered substantial acclaim due to their remarkable capacity to deliver high-quality solutions to the challenging FCTP. These techniques demonstrate substantial potential in accelerating the resolution of this formidable problem. The central goal revolves around the exploration of groundbreaking solutions for the Fixed Load Transportation Problem, all while concurrently minimizing the time investment required to attain these optimal solutions. This undertaking necessitates the adept utilization of the Dragonfly algorithm, an algorithm inspired by natural processes, known for its adaptability and robustness in solving complex problems. The FCTP, functioning as an optimization problem, grapples with the multifaceted task of formulating distribution plans for products originating from multiple sources and destined for various endpoints. The overarching aspiration is to minimize overall transportation costs, a challenge that mandates meticulous considerations, including product availability at source locations and demand projections at destination points. The proposed methodology introduces an innovative approach tailored explicitly for addressing the Fixed Charge Transport Problem (FCTP) by harnessing the inherent capabilities of the Dragonfly algorithm. This adaptation of the algorithm's underlying processes is precisely engineered to handle large-scale FCTP instances, with the ultimate objective of unveiling solutions that have hitherto remained elusive. The numerical results stemming from our rigorous experiments unequivocally underscore the remarkable prowess of the Dragonfly algorithm in discovering novel and exceptionally efficient solutions. This demonstration unequivocally reaffirms its effectiveness in overcoming the inherent challenges posed by substantial FCTP instances. In summary, the research represents a significant leap forward in the domain of FCTP solution methodologies by seamlessly integrating the formidable capabilities of the Dragonfly algorithm into the problem-solving process. The insights and solutions presented in this article hold immense promise for significantly enhancing the efficiency and effectiveness of FCTP resolution, ultimately benefiting a broad spectrum of industries and logistics systems, and promising advancements in the optimization of transportation processes

On the pythagorean structure of the optimal transport for separable cost functions

In this paper, we study the optimal transport problem induced by two measures supported over two polish spaces, namely, X and Y , which are the product of n smaller polish spaces, that is, X={\Large{\times}}_{j=1}^n X_j and Y={\Large{\times}}_{j=1}^n Y_j . In particular, we focus on problems induced by a cost function c:X\times Y\to [0,+\infty) that is separable; i.e., c is such that c=c_1+\dots+c_n , where each c_j depends only on the couple (x_j,y_j) , and thus c_j:X_j\times Y_j\to [0,+\infty) . Noticeably, if X=Y=\mathbb{R}^n , this class of cost functions includes all the l_p^p costs. Our main result proves that the optimal transportation plan with respect to a separable cost function between two given measures can be expressed as the composition of n different lower-dimensional transports, one for each pair of coordinates (x_i,y_i) in X\times Y . This allows us to decompose the entire Wasserstein cost as the sum of n lower-dimensional Wasserstein costs and to prove that there always exists an optimal transportation plan whose random variable enjoys a conditional independence property with respect to its marginals. We then show that our formalism allows us to explicitly compute the optimal transportation plan between two probability measures when each measure has independent marginals. Finally, we focus on two specific frameworks. In the first one, the cost function is a separable distance, i.e., d=d_1+d_2 , where both d_1 and d_2 are distances themselves. In the second one, both measures are supported over \mathbb{R}^n and the cost function is of the form c(x,y)=h(\vert x_1-y_1\vert)+h(\vert x_2-y_2\vert) , where h is a convex function such that h(0)=0 .

A column generation approach for the lexicographic optimization of intra-hospital transports

AbstractOver the last few years, the efficient design of processes in hospitals and medical facilities has received more and more attention, particularly when the improvement of the processes is aimed at relieving the workload of medical staff. To this end, we have developed a method to determine optimal allocations of intra-hospital transports to hospital transport employees. When optimizing transport plans in hospitals, there are various optimization goals to strive for. Therefore, we used a lexicographic approach to solve this multi-criteria optimization problem. To calculate optimal transport plans in a sufficiently short computation time, we have decomposed the problem at hand with the Dantzig–Wolfe reformulation and solved the resulting pricing subproblem with an enumerative column generation approach. To improve the efficiency of the column generation process, we have investigated and implemented different pruning methods, dominance rules and a column reuse mechanism for the online setting of the application at hand. In an extensive computational study, we first evaluated the efficiency of the different pruning methods before we compared our solution approach with the standard branch-and-bound column search approach, classical column generation methods and the solution of an integrated MIP model solved by a commercial solver. Finally, we present performance indicators of the transport optimization tool, which was developed from our method and is now productively used in a German hospital.

ОПТИМИЗАЦИЯ ЛОГИСТИЧЕСКОЙ СХЕМЫ ДЛЯ ОБЕСПЕЧЕНИЯ РАЙОНОВ РЕСПУБЛИКИ ТАТАРСТАН УДОБРЕНИЯМИ ИЗ ОТХОДОВ ПТИЦЕВОДСТВА

The problem of poultry agroindustrial complex sector waste’s recycling, in particular, live activity’s waste products (poultry droppings), is considered. The technology of their utilization in gasification installation by airless, oxygenfree thermal decomposition to produce fuel gas and valuable mineral fertilizer - ash method using is proposed. The technology of waste (manure) gasification for its environmental friendliness due to the nitrogen absence at gasifying medium is known. Ash after unloading from the gasification plant can be used in agriculture as a mineral fertilizer with a high content of phosphorus and potassium. The ash, in turn, for a long time is stored and does not require special transportation conditions. The waste into a useful product’s conversion will allow poultry enterprises to eliminate environmental payments for negative environmental impacts. Poultry waste in areas that can be divided into three large centers is generated, the poultry waste gasification enterprises’ construction will be carried out and innovative technologies for its recycling will be introduced. Raw materials for large enterprises are supposed to be delivered by the cluster method from small ones. The product obtained after gasification must be known on the market and accessible through the transport’s network. A solution is also proposed to fertilizers imbalance’s eliminating on the Republic of Tatarstan districts example by transport routes optimizing from equidistant centers to all consumers organizing. The transport problem solving method to find the best schemes for fertilizers’ transporting to areas with the greatest deficit was used. A matrix of paired comparisons is presented, and the optimal transportation plan is calculated. The results show that the proposed routes and the transported fertilizers amount can the region’s situation with the fertilizers’ imbalance improving. This technique allows the regions’ agroindustrial sector due to the search and transportation to areas’ costs as to fertilization’s necessity optimizing.