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Scaling Algorithms for Unbalanced Transport Problems
TLDR
This article introduces a new class of fast algorithms to approx-imate variational problems involving unbalanced optimal transport, and shows how these methods can be used to solve unbalanced transport, unbalanced gradient flows, and to compute unbalanced barycenters.Abstract:
This article introduces a new class of fast algorithms to approx-imate variational problems involving unbalanced optimal transport. While classical optimal transport considers only normalized probability distributions, it is important for many applications to be able to compute some sort of re-laxed transportation between arbitrary positive measures. A generic class of such “unbalanced” optimal transport problems has been recently proposed by several authors. In this paper, we show how to extend the, now classical, entropic regularization scheme to these unbalanced problems. This gives rise to fast, highly parallelizable algorithms that operate by performing only diagonal scaling (i.e. pointwise multiplications) of the transportation couplings. They are generalizations of the celebrated Sinkhorn algorithm. We show how these methods can be used to solve unbalanced transport, unbalanced gradient flows, and to compute unbalanced barycenters. We showcase applications to 2-D shape modification, color transfer, and growth models.read more
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Computational Optimal Transport
Gabriel Peyré,Marco Cuturi +1 more
TL;DR: 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.
Journal ArticleDOI
Optimal-Transport Analysis of Single-Cell Gene Expression Identifies Developmental Trajectories in Reprogramming.
Geoffrey Schiebinger,Geoffrey Schiebinger,Jian Shu,Jian Shu,Marcin Tabaka,Brian Cleary,Brian Cleary,Vidya Subramanian,Aryeh Solomon,Joshua Gould,Siyan Liu,Siyan Liu,Stacie Lin,Stacie Lin,Peter Berube,Lia Lee,Jenny Chen,Jenny Chen,Justin Brumbaugh,Philippe Rigollet,Konrad Hochedlinger,Rudolf Jaenisch,Aviv Regev,Aviv Regev,Aviv Regev,Eric S. Lander,Eric S. Lander,Eric S. Lander +27 more
TL;DR: Waddington-OT is introduced, an approach for studying developmental time courses to infer ancestor-descendant fates and model the regulatory programs that underlie them that sheds light on the process and outcome of reprogramming and provides a framework applicable to diverse temporal processes in biology.
Proceedings Article
Wasserstein Adversarial Examples via Projected Sinkhorn Iterations
TL;DR: A new threat model for adversarial attacks based on the Wasserstein distance is proposed, which can successfully attack image classification models, and it is demonstrated that PGD-based adversarial training can improve this adversarial accuracy to 76%.
Journal ArticleDOI
Convergence of Entropic Schemes for Optimal Transport and Gradient Flows
TL;DR: In the special case of the optimal transport problem, this technique dates back to the early 1970s as mentioned in this paper and was used to approximate solutions of linear programs in the early 1990s.
Posted Content
Stabilized Sparse Scaling Algorithms for Entropy Regularized Transport Problems
TL;DR: In this paper, a coarse-to-fine scaling algorithm for entropic transport-type problems has been proposed, which combines several modifications: a log-domain stabilized formulation, the well-known epsilon-scaling heuristic, an adaptive truncation of the kernel and a coarse to fine scheme.
References
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Book
Optimal Transport: Old and New
TL;DR: In this paper, the authors provide a detailed description of the basic properties of optimal transport, including cyclical monotonicity and Kantorovich duality, and three examples of coupling techniques.
Book
Topics in Optimal Transportation
TL;DR: In this paper, the metric side of optimal transportation is considered from a differential point of view on optimal transportation, and the Kantorovich duality of the optimal transportation problem is investigated.
Journal ArticleDOI
The Earth Mover's Distance as a Metric for Image Retrieval
TL;DR: This paper investigates the properties of a metric between two distributions, the Earth Mover's Distance (EMD), for content-based image retrieval, and compares the retrieval performance of the EMD with that of other distances.
Book
Gradient Flows: In Metric Spaces and in the Space of Probability Measures
TL;DR: In this article, Gradient flows and curves of Maximal slopes of the Wasserstein distance along geodesics are used to measure the optimal transportation problem in the space of probability measures.
Proceedings Article
Sinkhorn Distances: Lightspeed Computation of Optimal Transport
TL;DR: This work smooths the classic optimal transport problem with an entropic regularization term, and shows that the resulting optimum is also a distance which can be computed through Sinkhorn's matrix scaling algorithm at a speed that is several orders of magnitude faster than that of transport solvers.