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

Convex Analysis: (pms-28)

Optimal Transport: A Semi-Discrete Approach

II. Eignungsprüfung § 5 Inhaltliche Anforderungen und Durchführung der Eignungsprüfung § 6 Prüfungsausschuss § 7 Prüfungskommission § 8 Bewertung der Prüfungsleistungen § 9 Anrechnung anderer Leistungen § 10 Wiederholung der Prüfung § 11 Zuteilung freier Studienplätze § 12 Rücktritt, Ausschluss von der Prüfung, Rücknahme von Zulassungsund Prüfungsbescheiden § 13 Zeitliche Begrenzung der Zulassung und Immatrikulation

근첨하 분절 골절단술을 병행한 iii급 양악 전돌증의 교정치료 증례

In recent years the theory of the Wasserstein metric has opened up new treatments of diffusion equations as gradient systems, where the free energy or entropy take the role of the driving functional and where the space is equipped with the Wasserstein metric. We show on the formal level that this gradient structure can be generalized to reaction–diffusion systems with reversible mass-action kinetic. The metric is constructed using the dual dissipation potential, which is a quadratic functional of all chemical potentials including the mobilities as well as the reaction kinetics. The metric structure is obtained by Legendre transform from the dual dissipation potential.The same ideas extend to systems including electrostatic interactions or a correct energy balance via coupling to the heat equation. We show this by treating the semiconductor equations involving the electron and hole densities, the electrostatic potential, and the temperature. Thus, the models in Albinus et al (2002 Nonlinearity 15 367–83), which stimulated this work, have a gradient structure.

A gradient structure for reaction-diffusion systems and for energy-drift-diffusion systems

Nonparametric two-sample or homogeneity testing is a decision theoretic problem that involves identifying differences between two random variables without making parametric assumptions about their underlying distributions. The literature is old and rich, with a wide variety of statistics having being designed and analyzed, both for the unidimensional and the multivariate setting. In this short survey, we focus on test statistics that involve the Wasserstein distance. Using an entropic smoothing of the Wasserstein distance, we connect these to very different tests including multivariate methods involving energy statistics and kernel based maximum mean discrepancy and univariate methods like the Kolmogorov–Smirnov test, probability or quantile (PP/QQ) plots and receiver operating characteristic or ordinal dominance (ROC/ODC) curves. Some observations are implicit in the literature, while others seem to have not been noticed thus far. Given nonparametric two-sample testing’s classical and continued importance, we aim to provide useful connections for theorists and practitioners familiar with one subset of methods but not others.

On Wasserstein Two-Sample Testing and Related Families of Nonparametric Tests

We study the convergence of the graph Laplacian of a random geometric graph generated by an i.i.d. sample from a m-dimensional submanifold 
$${\mathcal {M}}$$

 in 
$$\mathbb {R}^d$$

 as the sample size n increases and the neighborhood size h tends to zero. We show that eigenvalues and eigenvectors of the graph Laplacian converge with a rate of 
$$O\Big (\big (\frac{\log n}{n}\big )^\frac{1}{2m}\Big )$$

 to the eigenvalues and eigenfunctions of the weighted Laplace–Beltrami operator of 
$${\mathcal {M}}$$

. No information on the submanifold 
$${\mathcal {M}}$$

 is needed in the construction of the graph or the “out-of-sample extension” of the eigenvectors. Of independent interest is a generalization of the rate of convergence of empirical measures on submanifolds in 
$$\mathbb {R}^d$$

 in infinity transportation distance.

Error Estimates for Spectral Convergence of the Graph Laplacian on Random Geometric Graphs Toward the Laplace–Beltrami Operator

A variational approach to the consistency of spectral clustering

We consider point clouds obtained as random samples of a measure on a Euclidean domain. A graph representing the point cloud is obtained by assigning weights to edges based on the distance between the points they connect. Our goal is to develop mathematical tools needed to study the consistency, as the number of available data points increases, of graph-based machine learning algorithms for tasks such as clustering. In particular, we study when the cut capacity, and more generally total variation, on these graphs is a good approximation of the perimeter (total variation) in the continuum setting. We address this question in the setting of Γ-convergence. We obtain almost optimal conditions on the scaling, as the number of points increases, of the size of the neighborhood over which the points are connected by an edge for the Γ-convergence to hold. Taking of the limit is enabled by a transportation based metric which allows us to suitably compare functionals defined on different point clouds.

Continuum Limit of Total Variation on Point Clouds

We consider random i.i.d. samples of absolutely continuous measures on bounded connected domains. We prove an upper bound on the -transportation distance between the measure and the empirical measure of the sample. The bound is optimal in terms of scaling with the number of sample points.

Nicolás García Trillos

Papers

On Wasserstein Two-Sample Testing and Related Families of Nonparametric Tests

Error Estimates for Spectral Convergence of the Graph Laplacian on Random Geometric Graphs Toward the Laplace–Beltrami Operator

A variational approach to the consistency of spectral clustering

Continuum Limit of Total Variation on Point Clouds

On the rate of convergence of empirical measures in ∞-transportation distance