Transport Map Research Articles

Պատերազմի տնտեսագիտություն. Ադրբեջանի կողմից նոր պատերազմ սկսելու տնտեսական դրդապատճառները/ ECONOMICS OF WAR: ECONOMIC MOTIVES FOR STARTING A NEW WAR BY AZERBAIJAN

The paper examines the complex motives behind Azerbaijan's potential initiation of a new conflict with Armenia, focusing on economic factors. It details how economic interests, particularly control over resources and strategic transport routes, play a central role in modern conflicts, often outweighing traditional political or ideological reasons. Azerbaijan's interest in the Syunik region of Armenia is highlighted as a case study, emphasizing the area's rich mineral resources and strategic position on regional and global transport maps. The analysis suggests that beyond immediate economic gains, controlling Syunik could offer significant strategic advantages to Azerbaijan, including enhanced trade efficiency and geopolitical leverage.

Limit theorems for entropic optimal transport maps and Sinkhorn divergence

Limit theorems for entropic optimal transport maps and Sinkhorn divergence

Uniform confidence band for optimal transport map on one-dimensional data

Uniform confidence band for optimal transport map on one-dimensional data

Towards optimal transport for quantum densities

An analogue of the quadratic Wasserstein (or Monge-Kantorovich) distance between Borel probability measures on $\mathbf{R}^d$ has been defined in [F. Golse, C. Mouhot, T. Paul: Commun. Math. Phys. 343 (2015), 165-205] for density operators on $L^2(\mathbf{R}^d)$, and used to estimate the convergence rate of various asymptotic theories in the context of quantum mechanics. The present work proves a Kantorovich type duality theorem for this quantum variant of the Monge-Kantorovich or Wasserstein distance, and discusses the structure of optimal quantum couplings. Specifically, we prove that optimal quantum couplings involve a gradient type structure similar to the Brenier transport map (which is the gradient of a convex function), or more generally, to the subdifferential of a l.s.c. convex function as in the Knott-Smith optimality criterion (see Theorem 2.12 in [C. Villani: Topics in Optimal Transportation, Amer. Math. Soc. 2003]).

Quantitative Transport Mapping Derived Tracer Velocity for Alzheimer’s Disease

AbstractBackgroundThe Alzheimer’s disease (AD), featured with extracellular beta‐amyloid plaque and intracellular tau tangles, are considered caused by the brain clearance function deficits. One anatomical representation of the fluid stasis is the enlargement of the perivascular space. Our hypothesis is that the fluid stasis will reduce the fluid velocity, and the reduced fluid velocity will drive the PET tracer slower than normal. We used the velocity map from our proposed quantitative transport mapping (QTM, L. Zhou MRM 2020.) to evaluate the PET tracer velocity in Alzheimer’s disease.MethodFifty‐seven subjects (age: 69.7±9.1, Female/male = 37/20, NL/MCI = 45/12) went through both 11C‐PIB and 18F‐MK‐6240 PET scan for Aβ and tau deposit measurements, and MRI scan for ROI parcellation. PIB SUVR, MK vCSF‐Slope (Y. Li Fluids Barriers CNS 2022), MK QTM are processed for the following analysis. Specifically, vCSF‐Slope was computed as the tracer turnover rate in lateral ventricle using the dynamic MK data between 10 to 30min, normalized by the whole brain AUC in 1 to 4min. QTM velocity was processed using the dynamic MK data between 40 to 55min using mass conservation equation. Regression model for predicting Aβ using QTM was conducted by controlling for age and sex. The relationship between QTM and vCSF was evaluated. The use of QTM to distinguish subjects groups are tested.ResultOur preliminary results showed that quantitative transport mapping(QTM) based on fluid mechanics is significantly associate the ventricular CSF clearance(vCSF) measured by dynamic PET data and brain amyloid(PiB PET). The regression model shows that cerebral cortex (GM) QTM velocity (p<0.01, t = ‐2.76) and age (p<0.05, t = 2.55) are significantly associated with the GM PIB SUVR as shown in Figure 1. This indicates that high Aβ deposit is associated with lower QTM velocity, i.e., clearance deficits. Figure 2 shows that in GM QTM velocity is correlated with the vCSF‐Slope (p<0.05, r = 0.37). Figure 3 shows that QTM velocity in both GM and lateral ventricle are able to distinguish normal and MCI/AD groups.ConclusionThe tracer velocity in the brain tissue measured by QTM is associated with ventricular CSF turnover rate and brain amyloid burden. Acknowledgement: This study is supported by NIA grant AG057848

On the Representation and Learning of Monotone Triangular Transport Maps

On the Representation and Learning of Monotone Triangular Transport Maps

Ensemble transport smoothing. Part II: Nonlinear updates

Smoothing is a specialized form of Bayesian inference for state-space models that characterizes the posterior distribution of a collection of states given an associated sequence of observations. Ramgraber et al. [38] proposes a general framework for transport-based ensemble smoothing, which includes linear Kalman-type smoothers as special cases. Here, we build on this foundation to realize and demonstrate nonlinear backward ensemble transport smoothers. We discuss parameterization and regularization of the associated transport maps, and then examine the performance of these smoothers for nonlinear and chaotic dynamical systems that exhibit non-Gaussian behavior. In these settings, our nonlinear transport smoothers yield lower estimation error than conventional linear smoothers and state-of-the-art iterative ensemble Kalman smoothers, for comparable numbers of model evaluations.

Ensemble transport smoothing. Part I: Unified framework

Smoothers are algorithms for Bayesian time series re-analysis. Most operational smoothers rely either on affine Kalman-type transformations or on sequential importance sampling. These strategies occupy opposite ends of a spectrum that trades computational efficiency and scalability for statistical generality and consistency: non-Gaussianity renders affine Kalman updates inconsistent with the true Bayesian solution, while the ensemble size required for successful importance sampling can be prohibitive. This paper revisits the smoothing problem from the perspective of measure transport, which offers the prospect of consistent prior-to-posterior transformations for Bayesian inference. We leverage this capacity by proposing a general ensemble framework for transport-based smoothing. Within this framework, we derive a comprehensive set of smoothing recursions based on nonlinear transport maps and detail how they exploit the structure of state-space models in fully non-Gaussian settings. We also describe how many standard Kalman-type smoothing algorithms emerge as special cases of our framework. A companion paper [35] explores the implementation of nonlinear ensemble transport smoothers in greater depth.

Optimal transport for some symmetric, multidimensional integer partitions

A result of Hohloch links the theory of integer partitions with the Monge formulation of the optimal transport problem, giving the optimal transport map between (Young diagrams of) integer partitions and their corresponding symmetric partitions. Our aim is to extend Hohloch’s result to the higher dimensional case. In doing so, we show the Kantorovich formulation of the optimal transport problem provides the tool to study the matching of higher dimensional partitions with their corresponding symmetric partitions.

Computational optimal transport for molecular spectra: The fully continuous case.

Computational optimal transport is used to analyze the difference between pairs of continuous molecular spectra. It is demonstrated that transport distances which are derived from this approach may be a more appropriate measure of the difference between two continuous spectra than more familiar measures of distance under many common circumstances. Associated with the transport distances is the transport map which provides a detailed analysis of the difference between two molecular spectra and is a key component of our study of quantitative differences between two continuous spectra. The use of optimal transport for comparing molecular spectra is developed in detail here with a set of model spectra, so that the discussion is self-contained. The difference between the transport distance and more common definitions of distance is elucidated for some well-chosen examples and it is shown where transport distances may be very useful alternatives to standard definitions of distance. The transport distance between a theoretical and experimental electronic absorption spectrum for SO2 is studied and it is shown how the theoretical spectrum can be modified to fit the experimental spectrum better adjusting the theoretical band origin and the resolution of the theoretical spectrum. This analysis includes the calculation of transport maps between the theoretical and experimental spectra suggesting future applications of the methodology.

Ultrawideband coupled relative positioning algorithm applicable to flight controller for multidrone collaboration

AbstractIn this study, we introduce a loosely coupled relative position estimation method that utilizes a decentralized ultrawideband (UWB), Global Navigation Support System and inertial navigation system for flight controllers (FCs). Key obstacles to multidrone collaboration include relative position errors and the absence of communication devices. To address this, we provide an extended Kalman filter‐based algorithm and module that correct distance errors by fusing UWB data acquired through random communications. Via simulations, we confirm the feasibility of the algorithm and verify its distance error correction performance according to the amount of communications. Real‐world tests confirm the algorithm's effectiveness on FCs and the potential for multidrone collaboration in real environments. This method can be used to correct relative multidrone positions during collaborative transportation and simultaneous localization and mapping applications.

Extended ray-mapping method based on differentiable ray-tracing for non-paraxial and off-axis freeform illumination lens design.

The ray-mapping method has been widely used for designing freeform illumination lenses. However, in non-paraxial or off-axis situations, it remains challenging to obtain an integrable ray-mapping, often requiring a complex iterative correction process for the initial mapping. To address this challenge, we propose an extended ray-mapping method that incorporates differentiable ray-tracing into the design pipeline of the ray-mapping method. This enables accurate surface construction according to ray-mapping and efficient shape correction based on irradiance distribution. The proposed method involves two optimization stages. In the first stage, the freeform surface is preliminarily optimized to closely match the optimal transport mapping. The obtained freeform surface is then further optimized in the second stage to minimize the divergence between the target and simulated irradiance distributions. Additionally, the mean curvature of the freeform surface is also constrained in the second stage to facilitate the fabrication of the final freeform surface. Non-paraxial illumination lenses and off-axis illumination lenses have been designed using the proposed method within ten minutes, and simulations demonstrate that the approach is effective and robust.

Behavior of the Poincaré constant along the Polchinski renormalization flow

We control the behavior of the Poincaré constant along the Polchinski renormalization flow using a dynamic version of [Formula: see text]-calculus. We also treat the case of higher order eigenvalues. Our method generalizes a method introduced by Klartag and Putterman to analyze the evolution of log-concave distributions along the heat flow. Furthermore, we apply it to general [Formula: see text]-measures and discuss the interpretation in terms of transport maps.

Probability flow solution of the Fokker–Planck equation

The method of choice for integrating the time-dependent Fokker–Planck equation (FPE) in high-dimension is to generate samples from the solution via integration of the associated stochastic differential equation (SDE). Here, we study an alternative scheme based on integrating an ordinary differential equation that describes the flow of probability. Acting as a transport map, this equation deterministically pushes samples from the initial density onto samples from the solution at any later time. Unlike integration of the stochastic dynamics, the method has the advantage of giving direct access to quantities that are challenging to estimate from trajectories alone, such as the probability current, the density itself, and its entropy. The probability flow equation depends on the gradient of the logarithm of the solution (its ‘score’), and so is a-priori unknown. To resolve this dependence, we model the score with a deep neural network that is learned on-the-fly by propagating a set of samples according to the instantaneous probability current. We show theoretically that the proposed approach controls the Kullback–Leibler (KL) divergence from the learned solution to the target, while learning on external samples from the SDE does not control either direction of the KL divergence. Empirically, we consider several high-dimensional FPEs from the physics of interacting particle systems. We find that the method accurately matches analytical solutions when they are available as well as moments computed via Monte-Carlo when they are not. Moreover, the method offers compelling predictions for the global entropy production rate that out-perform those obtained from learning on stochastic trajectories, and can effectively capture non-equilibrium steady-state probability currents over long time intervals.

Physically constrained covariance inflation from location uncertainty

Abstract. Motivated by the concept of “location uncertainty”, initially introduced in Mémin (2014), a scheme is sought to perturb the “location” of a state variable at every forecast time step. Further considering Brenier's theorem (Brenier, 1991), asserting that the difference of two positive density fields on the same domain can be represented by a transportation map, we demonstrate that the perturbations consistently define a stochastic partial differential equation (SPDE) from the original PDE. It ensues that certain quantities, up to the user, are conserved at every time step. Remarkably, derivations following both the SALT (stochastic advection by Lie transport; Holm, 2015) and LU (location uncertainty; Mémin, 2014; Resseguier et al., 2017a) settings can be recovered from this perturbation scheme. Still, it offers broader applicability since it does not explicitly rely on Lagrangian mechanics or Newton's laws of force. For illustration, a stochastic version of the thermal shallow water equation is presented.

“Commercial and political needs demand the establishment of a Black Sea port”: Infrastructure development, opportunities and anxieties in an Eastern European periphery (1860s–1870s)

This paper is the biography of a failed infrastructural project: that of building a Romanian seaport and of linking it to the mainland via a canal and/or a railway. It details its rich life, its initiators and partisans, but also its (mostly foreign) contenders; “born” in the early 1860s in a certain geopolitical, economic or infrastructural environment, the initiative morphed in the coming years in close connection and in direct competition with complementary or rival projects. While the plan of having an “independent” Romanian Black Sea port remained central to the Jibrieni project, most of its other infrastructural components were continuously reimagined, so as to fit into the Romanian government's priorities and – with them – into the extremely dynamic transportation map of Central and Eastern Europe in the 1860s–1870s.

Rendezvous Control Design for the Generalized Cucker–Smale Model on Riemannian Manifolds

In this paper, we consider a rendezvous problem for the generalized Cucker-Smale model, which is a double-integrator multi-agent system, on complete Riemannian manifolds. With the help of the covariant derivative, parallel transport and logarithm map on the Riemannian manifold, we design a rendezvous feedback law that enables all agents to converge at a given target in the Riemannian manifold, under some a priori conditions. Furthermore, we consider three concrete complete Riemannian manifolds, such as the unit circle, unit sphere, and hyperboloid, and present the explicit feedback laws for rendezvous on them by calculating the corresponding covariant derivatives, parallel transports, and logarithm maps. Meanwhile, numerical examples are given for the manifolds as mentioned above to verify and illustrate the theoretical results.

Efficient Multimodal Sampling via Tempered Distribution Flow

Sampling from high-dimensional distributions is a fundamental problem in statistical research and practice. However, great challenges emerge when the target density function is unnormalized and contains isolated modes. We tackle this difficulty by fitting an invertible transformation mapping, called a transport map, between a reference probability measure and the target distribution, so that sampling from the target distribution can be achieved by pushing forward a reference sample through the transport map. We theoretically analyze the limitations of existing transport-based sampling methods using the Wasserstein gradient flow theory, and propose a new method called TemperFlow that addresses the multimodality issue. TemperFlow adaptively learns a sequence of tempered distributions to progressively approach the target distribution, and we prove that it overcomes the limitations of existing methods. Various experiments demonstrate the superior performance of this novel sampler compared to traditional methods, and we show its applications in modern deep learning tasks such as image generation. The programming code for the numerical experiments is available in the supplementary material.

Autoregressive optimal transport models.

Series of univariate distributions indexed by equally spaced time points are ubiquitous in applications and their analysis constitutes one of the challenges of the emerging field of distributional data analysis. To quantify such distributional time series, we propose a class of intrinsic autoregressive models that operate in the space of optimal transport maps. The autoregressive transport models that we introduce here are based on regressing optimal transport maps on each other, where predictors can be transport maps from an overall barycenter to a current distribution or transport maps between past consecutive distributions of the distributional time series. Autoregressive transport models and their associated distributional regression models specify the link between predictor and response transport maps by moving along geodesics in Wasserstein space. These models emerge as natural extensions of the classical autoregressive models in Euclidean space. Unique stationary solutions of autoregressive transport models are shown to exist under a geometric moment contraction condition of Wu & Shao [(2004) Limit theorems for iterated random functions. Journal of Applied Probability 41, 425-436)], using properties of iterated random functions. We also discuss an extension to a varying coefficient model for first-order autoregressive transport models. In addition to simulations, the proposed models are illustrated with distributional time series of house prices across U.S. counties and annual summer temperature distributions.

Transport map Bayesian parameter estimation for dynamical systems

AbstractAccurate online state and parameter estimation of uncertain non‐linear dynamical systems is a demanding task that has been traditionally handled by adopting non‐linear Kalman Filters or particle filters. However, in case of Kalman filters the system needs to be linearised and for particle filters the computational demand can be high. Recent advances in optimal transport theory and the application to Bayesian model updating pave the way for other approaches to system and parameter identification. They also provide a way of formulating the problem in such a way that efficient online estimation for complex systems is possible. In this work, we investigate the properties of the transport map approach when compared to standard Markov Chain Monte Carlo in an off‐line setting as a first step towards on‐line parameter estimation. We apply both approaches to an analytical exponential model and a dynamical system with seven unknown parameters subjected to ground displacement. Details on the theory of transport maps and on the used MCMC algorithm are also given.