Gabriel Peyré and Marco Cuturi, Computational Optimal Transport, Foundations and Trends in Machine Learning, vol. 11, no. 5-6, pp. 355-607, 2019.
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.François-Pierre Paty, Alexandre d'Aspremont, and Marco Cuturi, Regularity as Regularization: Smooth and Strongly Convex Brenier Potentials in Optimal Transport, arXiv:1905.10812.
Building on these two formulations we propose algorithms to estimate and evaluate transport maps with desired regularity properties, benchmark their statistical performance, apply them to domain adaptation and visualize their action on a color transfer task.Applications:
Optimal transport (OT) has found practical applications in areas as diverse as supervised machine learning [24, 1, 13], graphics [38, 8], generative models [6, 35], NLP [25, 3], biology [26, 37] or imaging [33, 15].QCQP (quadratically constrained quadratic program) in Theorem 1:
For multivariate measures the problem boils down to solving alternatively a convex QCQP and a discrete OT problem.Color Transfer:
Given a source and a target image, the goal of color transfer is to transform the colors of the source image so that it looks similar to the target image color palette.
where
The value W corresponds to the Wasserstein distance between the color distribution of the image and the color distribution of Van Gogh’s portrait. The smaller W, the greater the fidelity to Van Gogh’s portrait colors. Smaller values of L give more uniform colors, while larger values of l give more contrast.
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