Optimal Transport Theory Research Articles

Selection of OAM signal constellations for atmospheric channels using optimal transport theory

Selection of OAM signal constellations for atmospheric channels using optimal transport theory

A Comprehensive Method for Example-Based Color Transfer with Holistic–Local Balancing and Unit-Wise Riemannian Information Gradient Acceleration

Color transfer, an essential technique in image editing, has recently received significant attention. However, achieving a balance between holistic color style transfer and local detail refinement remains a challenging task. This paper proposes an innovative color transfer method, named BHL, which stands for Balanced consideration of both Holistic transformation and Local refinement. The BHL method employs a statistical framework to address the challenge of achieving a balance between holistic color transfer and the preservation of fine details during the color transfer process. Holistic color transformation is achieved using optimal transport theory within the generalized Gaussian modeling framework. The local refinement module adjusts color and texture details on a per-pixel basis using a Gaussian Mixture Model (GMM). To address the high computational complexity inherent in complex statistical modeling, a parameter estimation method called the unit-wise Riemannian information gradient (uRIG) method is introduced. The uRIG method significantly reduces the computational burden through the second-order acceleration effect of the Fisher information metric. Comprehensive experiments demonstrate that the BHL method outperforms state-of-the-art techniques in both visual quality and objective evaluation criteria, even under stringent time constraints. Remarkably, the BHL method processes high-resolution images in an average of 4.874 s, achieving the fastest processing time compared to the baselines. The BHL method represents a significant advancement in the field of color transfer, offering a balanced approach that combines holistic transformation and local refinement while maintaining efficiency and high visual quality.

An optimal transport approach for 3D electrical impedance tomography

Abstract This work solves the three-dimensional inverse boundary value problem with the quadratic Wasserstein distance (W2), which originates from the optimal transportation (OT) theory. The computation of the W2 distance on the manifold surface is boiled down to solving the generalized Monge-Ampère equation, whose solution is directly related to the gradient of the W2 distance. An efficient first-order method based on iteratively solving Poisson's equation is introduced to solve the fully nonlinear elliptic equation. Combining with the adjoint-state technique, the optimization framework based on the W2 distance is developed to solve the three-dimensional electrical impedance tomography (EIT) problem. The proposed method is especially suitable for severely ill-posed and highly nonlinear inverse problems. Numerical experiments demonstrate that our method improves the stability and outperforms the traditional regularization methods.

Wasserstein task embedding for measuring task similarities

Measuring similarities between different tasks is critical in a broad spectrum of machine learning problems, including transfer, multi-task, continual, and meta-learning. Most current approaches to measuring task similarities are architecture-dependent: (1) relying on pre-trained models, or (2) training networks on tasks and using forward transfer as a proxy for task similarity. In this paper, we leverage the optimal transport theory and define a novel task embedding for supervised classification that is model-agnostic, training-free, and capable of handling (partially) disjoint label sets. In short, given a dataset with ground-truth labels, we perform a label embedding through multi-dimensional scaling and concatenate dataset samples with their corresponding label embeddings. Then, we define the distance between two datasets as the 2-Wasserstein distance between their updated samples. Lastly, we leverage the 2-Wasserstein embedding framework to embed tasks into a vector space in which the Euclidean distance between the embedded points approximates the proposed 2-Wasserstein distance between tasks. We show that the proposed embedding leads to a significantly faster comparison of tasks compared to related approaches like the Optimal Transport Dataset Distance (OTDD). Furthermore, we demonstrate the effectiveness of our embedding through various numerical experiments and show statistically significant correlations between our proposed distance and the forward and backward transfer among tasks on a wide variety of image recognition datasets.

EEG microstate transition cost correlates with task demands.

The ability to solve complex tasks relies on the adaptive changes occurring in the spatio-temporal organization of brain activity under different conditions. Altered flexibility in these dynamics can lead to impaired cognitive performance, manifesting for instance as difficulties in attention regulation, distraction inhibition, and behavioral adaptation. Such impairments result in decreased efficiency and increased effort in accomplishing goal-directed tasks. Therefore, developing quantitative measures that can directly assess the effort involved in these transitions using neural data is of paramount importance. In this study, we propose a framework to associate cognitive effort during the performance of tasks with electroencephalography (EEG) activation patterns. The methodology relies on the identification of discrete dynamical states (EEG microstates) and optimal transport theory. To validate the effectiveness of this framework, we apply it to a dataset collected during a spatial version of the Stroop task, a cognitive test in which participants respond to one aspect of a stimulus while ignoring another, often conflicting, aspect. The Stroop task is a cognitive test where participants must respond to one aspect of a stimulus while ignoring another, often conflicting, aspect. Our findings reveal an increased cost linked to cognitive effort, thus confirming the framework's effectiveness in capturing and quantifying cognitive transitions. By utilizing a fully data-driven method, this research opens up fresh perspectives for physiologically describing cognitive effort within the brain.

Anchor Space Optimal Transport as a Fast Solution to Multiple Optimal Transport Problems.

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.

Reliable underwater multi-target direction of arrival estimation with optimal transport using deep models.

Multi-target direction of arrival (DoA) estimation is an important and challenging task for sonar signal processing. In this study, we propose a method called learning direction of arrival with optimal transport (LOT) to accurately estimate the DoAs of multiple sources with a single deep model. We model the DoA estimation problem as a multi-label classification task and introduce an optimal transport (OT) loss based on the OT theory to capture the intrinsic continuity within the angular categories. We design a cost matrix for the OT loss in LOT approach to characterize the order and periodicity of the angular grid. The LOT approach encourages reliable predictions closer to the ground truth and suppresses spurious targets. We also propose a lightweight channel mask data augmentation module for deep models that use items related to the covariance matrix as input. The proposed methods can be seamlessly integrated with different model architectures and we indicate the portability with experiments on several typical network backbones. Experiments across various scenarios using different measurements show the effectiveness and robustness of our approaches. Results on SwellEx-96 experimental data demonstrate the practicality in real applications.

Applications of Optimal Transportation

The mathematical theory of optimal transportation is constantly expanding its range of application, while applications give impulses for new research directions in the field. This workshop was specifically devoted to applications of optimal transportation in the natural sciences, and to the recent developments of the theory that have been motivated by these. The bouquet of current applications includes mathematical models for large-scale air motion, dynamics of plasmas, material design, pattern formation in fluids, collective behaviour in biology, and many more. Related theoretical developments are in the analysis of the Hellinger-Kantorovich metric for modeling reaction–diffusion processes, and in efficient numerical methods for multi-marginal optimal transport, to name two of many examples encountered in this workshop.

Optimal light cone for macroscopic particle transport in long-range systems: A quantum speed limit approach

Understanding the ultimate rate at which information propagates is a pivotal issue in nonequilibrium physics. Nevertheless, the task of elucidating the propagation speed inherent in quantum bosonic systems presents challenges due to the unbounded nature of their interactions. In this study, we tackle the problem of macroscopic particle transport in a long-range generalization of the lattice Bose-Hubbard model through the lens of the quantum speed limit. By developing a unified approach based on optimal transport theory, we rigorously prove that the minimum time required for macroscopic particle transport is always bounded by the distance between the source and target regions, while retaining its significance even in the thermodynamic limit. Furthermore, we derive an upper bound for the probability of observing a specific number of bosons inside the target region, thereby providing additional insights into the dynamics of particle transport. Our results hold true for arbitrary initial states under both long-range hopping and long-range interactions, thus resolving an open problem of particle transport in generic bosonic systems.

A new perspective on denoising based on optimal transport

Abstract In the standard formulation of the classical denoising problem, one is given a probabilistic model relating a latent variable $\varTheta \in \varOmega \subset{\mathbb{R}}^{m} \; (m\ge 1)$ and an observation $Z \in{\mathbb{R}}^{d}$ according to $Z \mid \varTheta \sim p(\cdot \mid \varTheta )$ and $\varTheta \sim G^{*}$, and the goal is to construct a map to recover the latent variable from the observation. The posterior mean, a natural candidate for estimating $\varTheta $ from $Z$, attains the minimum Bayes risk (under the squared error loss) but at the expense of over-shrinking the $Z$, and in general may fail to capture the geometric features of the prior distribution $G^{*}$ (e.g. low dimensionality, discreteness, sparsity). To rectify these drawbacks, in this paper we take a new perspective on this denoising problem that is inspired by optimal transport (OT) theory and use it to study a different, OT-based, denoiser at the population level setting. We rigorously prove that, under general assumptions on the model, this OT-based denoiser is mathematically well-defined and unique, and is closely connected to the solution to a Monge OT problem. We then prove that, under appropriate identifiability assumptions on the model, the OT-based denoiser can be recovered solely from information of the marginal distribution of $Z$ and the posterior mean of the model, after solving a linear relaxation problem over a suitable space of couplings that is reminiscent of standard multimarginal OT problems. In particular, due to Tweedie’s formula, when the likelihood model $\{ p(\cdot \mid \theta ) \}_{\theta \in \varOmega }$ is an exponential family of distributions, the OT-based denoiser can be recovered solely from the marginal distribution of $Z$. In general, our family of OT-like relaxations is of interest in its own right and for the denoising problem suggests alternative numerical methods inspired by the rich literature on computational OT.

Similarity and economy of scale in urban transportation networks and optimal transport-based infrastructures

Designing and optimizing the structure of urban transportation networks is a challenging task. In this study, we propose a method inspired by optimal transport theory and the principle of economy of scale that uses little information in input to generate structures that are similar to those of public transportation networks. Contrarily to standard approaches, it does not assume any initial backbone network infrastructure but rather extracts this directly from a continuous space using only a few origin and destination points, generating networks from scratch. Analyzing a set of urban train, tram and subway networks, we find a noteworthy degree of similarity in several of the studied cases between simulated and real infrastructures. By tuning one parameter, our method can simulate a range of different subway, tram and train networks that can be further used to suggest possible improvements in terms of relevant transportation properties. Outputs of our algorithm provide naturally a principled quantitative measure of similarity between two networks that can be used to automatize the selection of similar simulated networks.

Thermodynamically optimal information gain in finite-time measurement

The tradeoff relation between speed and cost is a central issue in designing fast and efficient information processing devices. We derive an achievable bound on thermodynamic cost for obtaining information through finite-time (non-quasi-static) measurements. Our proof is based on optimal transport theory, which enables us to identify the explicit protocol to achieve the obtained bound. Moreover, we demonstrate that the optimal protocol can be approximately implemented by an experimentally feasible setup with quantum dots. Our results would lead to design principles of high-speed and low-energy-cost information processing. Published by the American Physical Society 2024

Continual Learning of Generative Models With Limited Data: From Wasserstein-1 Barycenter to Adaptive Coalescence.

Learning generative models is challenging for a network edge node with limited data and computing power. Since tasks in similar environments share a model similarity, it is plausible to leverage pretrained generative models from other edge nodes. Appealing to optimal transport theory tailored toward Wasserstein-1 generative adversarial networks (WGANs), this study aims to develop a framework that systematically optimizes continual learning of generative models using local data at the edge node while exploiting adaptive coalescence of pretrained generative models. Specifically, by treating the knowledge transfer from other nodes as Wasserstein balls centered around their pretrained models, continual learning of generative models is cast as a constrained optimization problem, which is further reduced to a Wasserstein-1 barycenter problem. A two-stage approach is devised accordingly: 1) the barycenters among the pretrained models are computed offline, where displacement interpolation is used as the theoretic foundation for finding adaptive barycenters via a "recursive" WGAN configuration and 2) the barycenter computed offline is used as metamodel initialization for continual learning, and then, fast adaptation is carried out to find the generative model using the local samples at the target edge node. Finally, a weight ternarization method, based on joint optimization of weights and threshold for quantization, is developed to compress the generative model further. Extensive experimental studies corroborate the effectiveness of the proposed framework.

Global Sensitivity Analysis via Optimal Transport

We examine the construction of variable importance measures for multivariate responses using the theory of optimal transport. We start with the classical optimal transport formulation. We show that the resulting sensitivity indices are well-defined under input dependence, are equal to zero under statistical independence, and are maximal under fully functional dependence. Also, they satisfy a continuity property for information refinements. We show that the new indices encompass Wagner’s variance-based sensitivity measures. Moreover, they provide deeper insights into the effect of an input’s uncertainty, quantifying its impact on the output mean, variance, and higher-order moments. We then consider the entropic formulation of the optimal transport problem and show that the resulting global sensitivity measures satisfy the same properties, with the exception that, under statistical independence, they are minimal, but not necessarily equal to zero. We prove the consistency of a given-data estimation strategy and test the feasibility of algorithmic implementations based on alternative optimal transport solvers. Application to the assemble-to-order simulator reveals a significant difference in the key drivers of uncertainty between the case in which the quantity of interest is profit (univariate) or inventory (multivariate). The new importance measures contribute to meeting the increasing demand for methods that make black-box models more transparent to analysts and decision makers. This paper was accepted by Baris Ata, stochastic models and simulation. Funding: A. Figalli acknowledges the support of the ERC [Grant 721675] “Regularity and Stability in Partial Differential Equations (RSPDE)” and of the Lagrange Mathematics and Computation Research Center. Supplemental Material: The online appendix and data files are available at https://doi.org/10.1287/mnsc.2023.01796 .

Optimal Transport with Constraints: From Mirror Descent to Classical Mechanics.

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.

Velocity model-based adapted meshes using optimal transport

In geophysical numerical models using the finite-element method or its variant, the spectral-element method, to solve seismic wave equations, a mesh is employed to discretize the domain. Generating or adapting a mesh to complex geological properties is a challenging task. To tackle this challenge, we develop an r-adaptivity method to generate or adapt a two-dimensional mesh to a seismic velocity field. Our scheme relies on the optimal transport theory to perform vertices relocation, which generates good-shaped meshes and prevents tangled elements. The mesh adaptation can delineate different regions of interest, like sharp interfaces, salt bodies, and discontinuities. The algorithm has a few user-defined parameters that control the mesh density. With typical seismic velocity examples (e.g., Camembert, SEAM Phase, Marmousi-2), mesh adaptation capability is illustrated within meshes with triangular and quadrilateral elements, commonly employed in seismic codes. Besides its potential use in mesh generation, the method developed can be embedded in seismic inversion workflows like multiscale full waveform inversion to adapt the mesh to the field being inverted without incurring the I/O cost of re-meshing and load rebalancing in parallel computations. The method can be extended to three-dimensional meshes.

Wasserstein barycenter regression: application to the joint dynamics of regional GDP and life expectancy in Italy

AbstractWe propose to investigate the joint dynamics of regional gross domestic product and life expectancy in Italy through Wasserstein barycenter regression derived from optimal transport theory. Wasserstein barycenter regression has the advantage of being flexible in modeling complex data distributions, given its ability to capture multimodal relationships, while maintaining the possibility of incorporating uncertainty and priors, other than yielding interpretable results. The main findings reveal that regional clusters tend to emerge, highlighting inequalities in Italian regions in economic and life expectancy terms. This suggests that targeted policy actions at a regional level fostering equitable development, especially from an economic viewpoint, might reduce regional inequality. Our results are validated by a robustness check on a human mobility dataset and by an illustrative forecasting exercise, which confirms the model’s ability to estimate and predict joint distributions and produce novel empirical evidence.

Unbalanced Optimal Transport For Stochastic Particle Tracking

Non-invasive flow measurement techniques, such as particle tracking velocimetry, resolve 3D velocity fields by pairing tracer particle positions in successive time steps. These trajectories are crucial for evaluating physical quantities like vorticity, shear stress, pressure, and coherent structures. Traditional approaches deterministically reconstruct particle positions and extract particle tracks using tracking algorithms. However, reliable track estimation is challenging due to measurement noise caused by high particle density, particle image overlap, and falsely reconstructed 3D particle positions. To overcome this challenge, probabilistic approaches quantify the epistemic uncertainty in particle positions, typically using a Gaussian probability distribution. However, the standard deterministic tracking algorithms relying on nearest-neighbor search do not directly extend to the probabilistic setting. Moreover, such algorithms do not necessarily find globally consistent solutions robust to reconstruction errors. This paper aims to develop a globally consistent nearest-neighborhood algorithm that robustly extracts stochastic particle tracks from the reconstructed Gaussian particle distributions in all frames. Our tracking algorithm relies on the unbalanced optimal transport theory in the metric space of Gaussian measures. Specifically, we optimize a binary transport plan for efficiently moving the Gaussian distributions of reconstructed particle positions between time frames. We achieve this by computing the partial Wasserstein distance in the metric space of Gaussian measures. Our tracking algorithm is robust to position reconstruction errors since it automatically detects the number of particles that should be matched through hyperparameter optimization. Notably, our tracking algorithm also readily applies to the standard deterministic PTV case. Finally, we validate our method using an in vitro flow experiment using a 3D-printed cerebral aneurysm.

Convex integrals of molecules in Lipschitz-free spaces

We introduce convex integrals of molecules in Lipschitz-free spaces F(M) as a continuous counterpart of convex series considered elsewhere, based on the de Leeuw representation. Using optimal transport theory, we show that these elements are determined by cyclical monotonicity of their supports, and that under certain finiteness conditions they agree with elements of F(M) that are induced by Radon measures on M, or that can be decomposed into positive and negative parts. We also show that convex integrals differ in general from convex series of molecules. Finally, we present some standalone results regarding extensions of Lipschitz functions which, combined with the above, yield applications to the extremal structure of F(M). In particular, we show that all elements of F(M) are convex series of molecules when M is uniformly discrete and identify all extreme points of the unit ball of F(M) in that case.

Spatio-Temporal Graph Hubness Propagation Model for Dynamic Brain Network Classification.

Dynamic brain network has the advantage over static brain network in characterizing the variation pattern of functional brain connectivity, and it has attracted increasing attention in brain disease diagnosis. However, most of the existing dynamic brain networks analysis methods rely on extracting features from independent brain networks divided by sliding windows, making them hard to reveal the high-order dynamic evolution laws of functional brain networks. Additionally, they cannot effectively extract the spatio-temporal topology features in dynamic brain networks. In this paper, we propose to use optimal transport (OT) theory to capture the topology evolution of the dynamic brain networks, and develop a multi-channel spatio-temporal graph convolutional network that collaboratively extracts the temporal and spatial features from the evolution networks. Specifically, we first adaptively evaluate the graph hubness of brain regions in the brain network of each time window, which comprehensively models information transmission among multiple brain regions. Second, the hubness propagation information across adjacent time windows is captured by optimal transport, describing high-order topology evolution of dynamic brain networks. Moreover, we develop a spatio-temporal graph convolutional network with attention mechanism to collaboratively extract the intrinsic temporal and spatial topology information from the above networks. Finally, the multi-layer perceptron is adopted for classifying the dynamic brain network. The extensive experiment on the collected epilepsy dataset and the public ADNI dataset show that our proposed method not only outperforms several state-of-the-art methods in brain disease diagnosis, but also reveals the key dynamic alterations of brain connectivities between patients and healthy controls.