Optimal transport (OT) theory can be informally described using the words of the French mathematician Gaspard Monge (1746-1818): A worker with a shovel in hand has to move a large pile of sand lying on a construction site. The goal of the worker is to erect with all that sand a target pile with a prescribed shape (for example, that of a giant sand castle). Naturally, the worker wishes to minimize her total effort, quantified for instance as the total distance or time spent carrying shovelfuls of sand. Mathematicians interested in OT cast that problem as that of comparing two probability distributions, two different piles of sand of the same volume. They consider all of the many possible ways to morph, transport or reshape the first pile into the second, and associate a "global" cost to every such transport, using the "local" consideration of how much it costs to move a grain of sand from one place to another. Recent years have witnessed the spread of OT in several fields, thanks to the emergence of approximate solvers that can scale to sizes and dimensions that are relevant to data sciences. Thanks to this newfound scalability, OT is being increasingly used to unlock various problems in imaging sciences (such as color or texture processing), computer vision and graphics (for shape manipulation) or machine learning (for regression, classification and density fitting). This short book reviews OT with a bias toward numerical methods and their applications in data sciences, and sheds lights on the theoretical properties of OT that make it particularly useful for some of these applications.

Computational Optimal Transport

Optimal Transport: A Semi-Discrete Approach

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Nonlinear Functional Analysis And Its Applications

5分で分かる!? 有名論文ナナメ読み：Jacot, Arthor, Gabriel, Franck and Hongler, Clement : Neural Tangent Kernel : Convergence and Generalization in Neural Networks

Information geometry is a new mathematical discipline which applies the methodology of differential geometry to statistics. Therefore, families of exponential distributions are considered as embedded manifolds, called statistical manifolds. This includes so important families like the multivariate normal or the gamma distributions. Fisher information — well known in information theory — becomes a metric on statistical manifolds. The Fisher information metric enables a hyperbolic structure on the multivariate normal distributions. Information geometry offers new methods for hypothesis testings, estimation theory or stochastic filtering. These can be used in engineering areas like signal processing or video processing or finance.

http://ieeexplore.ieee.org/iel5/6423297/6450609/06450636.pdf

Information geometry and its applications

Solving inverse problems without the use of derivatives or adjoints of the forward model is highly desirable in many applications arising in science and engineering. In this paper we propose a new ...

/pdf/interacting-langevin-diffusions-gradient-structure-and-3x3er65qmd.pdf

Interacting Langevin Diffusions: Gradient Structure and Ensemble Kalman Sampler

Mean field games (MFG) and mean field control (MFC) are critical classes of multiagent models for the efficient analysis of massive populations of interacting agents. Their areas of application span topics in economics, finance, game theory, industrial engineering, crowd motion, and more. In this paper, we provide a flexible machine learning framework for the numerical solution of potential MFG and MFC models. State-of-the-art numerical methods for solving such problems utilize spatial discretization that leads to a curse of dimensionality. We approximately solve high-dimensional problems by combining Lagrangian and Eulerian viewpoints and leveraging recent advances from machine learning. More precisely, we work with a Lagrangian formulation of the problem and enforce the underlying Hamilton-Jacobi-Bellman (HJB) equation that is derived from the Eulerian formulation. Finally, a tailored neural network parameterization of the MFG/MFC solution helps us avoid any spatial discretization. Our numerical results include the approximate solution of 100-dimensional instances of optimal transport and crowd motion problems on a standard work station and a validation using a Eulerian solver in two dimensions. These results open the door to much-anticipated applications of MFG and MFC models that are beyond reach with existing numerical methods.

https://www.pnas.org/content/pnas/117/17/9183.full.pdf

A machine learning framework for solving high-dimensional mean field game and mean field control problems

We propose a new algorithm to approximate the Earth Mover’s distance (EMD). Our main idea is motivated by the theory of optimal transport, in which EMD can be reformulated as a familiar $$L_1$$
 type minimization. We use a regularization which gives us a unique solution for this $$L_1$$
 type problem. The new regularized minimization is very similar to problems which have been solved in the fields of compressed sensing and image processing, where several fast methods are available. In this paper, we adopt a primal-dual algorithm designed there, which uses very simple updates at each iteration and is shown to converge very rapidly. Several numerical examples are provided.

A Parallel Method for Earth Mover’s Distance

We study a natural Wasserstein gradient flow on manifolds of probability distributions with discrete sample spaces. We derive the Riemannian structure for the probability simplex from the dynamical formulation of the Wasserstein distance on a weighted graph. We pull back the geometric structure to the parameter space of any given probability model, which allows us to define a natural gradient flow there. In contrast to the natural Fisher–Rao gradient, the natural Wasserstein gradient incorporates a ground metric on sample space. We illustrate the analysis of elementary exponential family examples and demonstrate an application of the Wasserstein natural gradient to maximum likelihood estimation.

Natural gradient via optimal transport

We present a primal-dual method to solve $L^1$-type nonsmooth optimization problems independently of the grid size. We apply these results to two important problems: the Rudin--Osher--Fatemi image ...

Wuchen Li

Papers

Interacting Langevin Diffusions: Gradient Structure and Ensemble Kalman Sampler

A machine learning framework for solving high-dimensional mean field game and mean field control problems

A Parallel Method for Earth Mover’s Distance

Natural gradient via optimal transport

Solving Large-Scale Optimization Problems with a Convergence Rate Independent of Grid Size