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Computational Optimal Transport
TLDR
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.
Iterative Bregman Projections for Regularized Transportation Problems
TLDR
It is shown that for many problems related to optimal transport, the set of linear constraints can be split in an intersection of a few simple constraints, for which the projections can be computed in closed form.
Wasserstein Barycenter and Its Application to Texture Mixing
TLDR
A new definition of the averaging of discrete probability distributions as a barycenter over the Monge-Kantorovich optimal transport space is proposed and a new fast gradient descent algorithm is introduced to compute Wasserstein barycenters of point clouds.
Interpolating between Optimal Transport and MMD using Sinkhorn Divergences
TLDR
This paper studies the Sinkhorn Divergences, a family of geometric divergences that interpolates between MMD and OT, and provides theoretical guarantees for positivity, convexity and metrization of the convergence in law.
Learning Generative Models with Sinkhorn Divergences
TLDR
This paper presents the first tractable computational method to train large scale generative models using an optimal transport loss, and tackles three issues by relying on two key ideas: entropic smoothing, which turns the original OT loss into one that can be computed using Sinkhorn fixed point iterations; and algorithmic (automatic) differentiation of these iterations.
Gromov-Wasserstein Averaging of Kernel and Distance Matrices
TLDR
This paper presents a new technique for computing the barycenter of a set of distance or kernel matrices, which define the interrelationships between points sampled from individual domains, and provides a fast iterative algorithm for the resulting nonconvex optimization problem.
Sliced and Radon Wasserstein Barycenters of Measures
TLDR
Two approaches to compute barycenters of measures using 1-D Wasserstein distances along radial projections of the input measures using the Radon transform are detailed.
Stochastic Optimization for Large-scale Optimal Transport
TLDR
A new class of stochastic optimization algorithms to cope with large-scale problems routinely encountered in machine learning applications, based on entropic regularization of the primal OT problem, which results in a smooth dual optimization optimization which can be addressed with algorithms that have a provably faster convergence.
Computational Optimal Transport: With Applications to Data Science
TLDR
Computational Optimal Transport presents an overview of the main theoretical insights that support the practical effectiveness of OT before explaining how to turn these insights into fast computational schemes.
A Generalized Forward-Backward Splitting
TLDR
This paper introduces the generalized forward-backward splitting algorithm for minimizing convex functions of the form F + G_i, and proves its convergence in infinite dimension, and its robustness to errors on the computation of the proximity operators and of the gradient of $F$.
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