Optimal Transport Map Research Articles

Measure-Driven Neural Solver for Optimal Transport Mapping

Measure-Driven Neural Solver for Optimal Transport Mapping

A volumetric approach to Monge's optimal transport on surfaces

We propose a volumetric formulation for computing the Optimal Transport problem defined on surfaces in R3, found in disciplines like optics, computer graphics, and computational methodologies. Instead of directly tackling the original problem on the surface, we define a new Optimal Transport problem on a thin tubular region, Tϵ, adjacent to the surface. This extension offers enhanced flexibility and simplicity for numerical discretization on Cartesian grids. The Optimal Transport mapping and potential function computed on Tϵ are consistent with the original problem on surfaces. We demonstrate that, with the proposed volumetric approach, it is possible to use simple and straightforward numerical methods to solve Optimal Transport for Γ=S2 and the 2-torus.

Reducing Mode Collapse With Monge-Kantorovich Optimal Transport for Generative Adversarial Networks.

Mode collapse has been a persisting challenge in generative adversarial networks (GANs), and it directly affects the applications of GAN in many domains. Existing works that attempt to solve this problem have some serious limitations: models using optimal transport (OT) strategies (e.g., Wasserstein distance) lead to vanishing or exploding gradients; increasing the number of generators can cause several generators focusing on the same mode; and approaches that modify the loss also do not satisfactorily resolve mode collapse. In this article, we reduce mode collapse by formulating it as a Monge problem of OT map. We show that the Monge problem can be transformed to the distribution transformation problem in GAN, and a rectified affine neural network can be considered as a measurable function. In this way, we propose Monge GAN that uses this measurable function to transform the generated data distribution into the original data distribution. We utilize the Kantorovich formulation to obtain the OT cost, which is regarded as the OT distance between the two distributions. Finally, we conduct extensive experiments on both image and numerical datasets to validate our Monge GAN in reducing model collapse.

Quantitative Stability of the Pushforward Operation by an Optimal Transport Map

Quantitative Stability of the Pushforward Operation by an Optimal Transport Map

Optimal Transport Map With Prescribed Direction Indicator for Seismic Full‐Waveform Inversion

AbstractThe quadratic Wasserstein (W2) metric has been proposed as a promising misfit function to mitigate cycle‐skipping phenomena in full‐waveform inversion. Mathematically, we demonstrate that the smoothness of the W2‐based adjoint source is two orders of magnitude higher than that based on L2‐norm, which guarantees a larger convergence radius of related inverse problems. However, the oscillatory characteristics of seismic signals and subsequent operations of transforming them into probability densities would decrease the accuracy of the optimal transport map T(t) and exacerbate the nonconvexity of the misfit function. To tackle these challenges, we propose the concept of prescribed direction indicator, which indicates the properly matching direction from predictions to observations, in order to correct inaccurate T(t). 1D synthetic examples suggest that reasonable bijection can be constructed through the proposed method. Numerical experiments demonstrate that it works well during optimization procedures, including enlarging the convergence radius of the inverse problem, improving the computational efficiency and enhancing the reliability of inversion results.

Moment-SoS methods for optimal transport problems

Most common optimal transport (OT) solvers are currently based on an approximation of underlying measures by discrete measures. However, it is sometimes relevant to work only with moments of measures instead of the measure itself, and many common OT problems can be formulated as moment problems (the most relevant examples being Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^p$$\\end{document}-Wasserstein distances, barycenters, and Gromov–Wasserstein discrepancies on Euclidean spaces). We leverage this fact to develop a generalized moment formulation that covers these classes of OT problems. The transport plan is represented through its moments on a given basis, and the marginal constraints are expressed in terms of moment constraints. A practical computation then consists in considering a truncation of the involved moment sequences up to a certain order, and using the polynomial sums-of-squares hierarchy for measures supported on semi-algebraic sets. We prove that the strategy converges to the solution of the OT problem as the order increases. We also show how to approximate linear quantities of interest, and how to estimate the support of the optimal transport map from the computed moments using Christoffel–Darboux kernels. Numerical experiments illustrate the good behavior of the approach.

The Brownian transport map

Contraction properties of transport maps between probability measures play an important role in the theory of functional inequalities. The actual construction of such maps, however, is a non-trivial task and, so far, relies mostly on the theory of optimal transport. In this work, we take advantage of the infinite-dimensional nature of the Gaussian measure and construct a new transport map, based on the Föllmer process, which pushes forward the Wiener measure onto probability measures on Euclidean spaces. Utilizing the tools of the Malliavin and stochastic calculus in Wiener space, we show that this Brownian transport map is a contraction in various settings where the analogous questions for optimal transport maps are open. The contraction properties of the Brownian transport map enable us to prove functional inequalities in Euclidean spaces, which are either completely new or improve on current results. Further and related applications of our contraction results are the existence of Stein kernels with desirable properties (which lead to new central limit theorems), as well as new insights into the Kannan–Lovász–Simonovits conjecture. We go beyond the Euclidean setting and address the problem of contractions on the Wiener space itself. We show that optimal transport maps and causal optimal transport maps (which are related to Brownian transport maps) between the Wiener measure and other target measures on Wiener space exhibit very different behaviors.

On distributional autoregression and iterated transportation

We consider the problem of defining and fitting models of autoregressive time series of probability distributions on a compact interval of . An order‐1 autoregressive model in this context is to be understood as a Markov chain, where one specifies a certain structure (regression) for the one‐step conditional Fréchet mean with respect to a natural probability metric. We construct and explore different models based on iterated random function systems of optimal transport maps. While the properties and interpretation of these models depend on how they relate to the iterated transport system, they can all be analyzed theoretically in a unified way. We present such a theoretical analysis, including convergence rates, and illustrate our methodology using real and simulated data. Our approach generalizes or extends certain existing models of transportation‐based regression and autoregression, and in doing so also provides some additional insights on existing models.

Three-stage transfer learning for motor imagery EEG recognition.

Motor imagery (MI) paradigms have been widely used in neural rehabilitation and drowsiness state assessment. The progress in brain-computer interface (BCI) technology has emphasized the importance of accurately and efficiently detecting motor imagery intentions from electroencephalogram (EEG). Despite the recent breakthroughs made in developing EEG-based algorithms for decoding MI, the accuracy and efficiency of these models remain limited by technical challenges posed by cross-subject heterogeneity in EEG data processing and the scarcity of EEG data for training. Inspired by the optimal transport theory, this study aims to develop a novel three-stage transfer learning (TSTL) method, which uses the existing labeled data from a source domain to improve classification performance on an unlabeled target domain. Notably, the proposed method comprises three components, namely, the Riemannian tangent space mapping (RTSM), source domain transformer (SDT), and optimal subspace mapping (OSM). The RTSM maps a symmetric positive definite matrix from the Riemannian space to the tangent space to minimize the marginal probability distribution drift. The SDT transforms the source domain to a target domain by finding the optimal transport mapping matrix to reduce the joint probability distribution differences. The OSM finally maps the transformed source domain and original target domain to the same subspace to further mitigate the distribution discrepancy. The performance of the proposed method was validated on two public BCI datasets, and the average accuracy of the algorithm on two datasets was 72.24% and 69.29%. Our results demonstrated the improved performance of EEG-based MI detection in comparison with state-of-the-art algorithms.

Entropic optimal transport solutions of the semigeostrophic equations

The semigeostrophic equations are a frontogenesis model in atmospheric science. Existence of solutions both from the theoretical and numerical point of view is given under a change of variable involving the interpretation of the pressure gradient as an optimal transport map between the density of the fluid and its push forward. Thanks to recent advances in numerical optimal transportation, the computation of large scale discrete approximations can be envisioned. We study here the use of entropic optimal transport and its Sinkhorn Algorithm companion.

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

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.

Optimal transport-based unsupervised semantic disentanglement: A novel approach for efficient image editing in GANs

The latent space of pre-trained generative adversarial networks (GANs) is rich in semantic information, which often becomes highly entangled. It is crucial to identify semantic directions within this latent space, as these directions correlate with image attributes and are vital for image editing tasks. Existing methods for semantic discovery usually involve labor-intensive procedures such as manual labeling and training attribute classifiers, which limits their practicality. In response to this issue, the paper proposes the Optimal Transport-based Unsupervised Semantic Disentanglement (OTUSD) algorithm. This novel method efficiently uncovers semantic directions in the latent space of GANs by utilizing the concepts of manifold learning and optimal transport (OT) theory. OTUSD applies singular value decomposition (SVD) to the OT matrix that links latent codes to generated images. This process yields singular vectors that correspond to semantically meaningful directions. Unlike traditional methods, OTUSD bypasses the need for time-consuming labeling and training processes, thus enhancing efficiency and revealing a wider array of semantically meaningful directions. Experimental results demonstrate the effectiveness of OTUSD in discovering semantic directions from several state-of-the-art GAN models, including StyleGAN, StyleGAN2, and BigGAN. This performance emphasizes the potential applicability of OTUSD to image editing and other related tasks, and illuminates its value in harnessing the manifold learning and OT mapping capabilities inherent in GANs for semantic disentanglement. The implementation code is available at https://github.com/LuckAlex/OTUSD.

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.

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.

Second order derivative of a functional associated to an optimal transport map

Second order derivative of a functional associated to an optimal transport map

The $L^1$-contraction principle in optimal transport

In this work, we use the JKO scheme to approximate a general class of diffusion problems generated by Darcy's law. Although the scheme is now classical, if the energy density is spatially inhomogeneous or irregular, many standard methods fail to apply to establish convergence in the continuum limit. To overcome these difficulties, we analyze the scheme through its dual problem and establish a novel $L^1$-contraction principle for the density variable. Notably, the contraction principle relies only on the existence of an optimal transport map and the convexity structure of the energy. As a result, the principle holds in a very general setting, and opens the door to using optimal-transport-based variational schemes to study a larger class of non-linear inhomogeneous parabolic equations.

Minimax rate for optimal transport regression between distributions

Distribution-on-distribution regression considers the problem of formulating and estimating a regression relationship where both covariate and response are probability distributions. The optimal transport distributional regression model postulates that the conditional Fréchet mean of the response distribution is linked to the covariate distribution via an optimal transport map. We establish the minimax rate of estimation of such a regression function, by deriving a lower bound that matches the convergence rate attained by the Fréchet least squares estimator.

Optimal Transport for a Class of Linear Quadratic Differential Games

We consider a setting where two noncooperative players optimally influence the evolution of an initial spatial probability in a game-theoretic hierarchical fashion (Stackelberg differential game), so that at a specific final time the distribution of the state matches a given final target measure. We provide a sufficient condition for the existence and uniqueness of an optimal transport map and prove that it can be characterized as the gradient of some convex function. An important by-product of our formulation is that it provides a means to study a class of Stackelberg differential games where the initial and final states of the underlying system are uncertain, but drawn randomly from some probability measures.