# Optimal Transport to a Variety

@inproceedings{Celik2019OptimalTT,
title={Optimal Transport to a Variety},
author={Turku Ozlum Celik and Asgar Jamneshan and Guido Mont{\'u}far and Bernd Sturmfels and Lorenzo Venturello},
booktitle={MACIS},
year={2019}
}
We study the problem of minimizing the Wasserstein distance between a probability distribution and an algebraic variety. We consider the setting of finite state spaces and describe the solution depending on the choice of the ground metric and the given distribution. The Wasserstein distance between the distribution and the variety is the minimum of a linear functional over a union of transportation polytopes. We obtain a description in terms of the solutions of a finite number of systems of… Expand
7 Citations

#### Figures, Tables, and Topics from this paper

Tropical optimal transport and Wasserstein distances
• Mathematics
• 2019
We study the problem of optimal transport in tropical geometry and define the Wasserstein-$p$ distances for probability measures in the continuous metric measure space setting of the tropicalExpand
Tropical Optimal Transport and Wasserstein Distances in Phylogenetic Tree Space
• Mathematics
• 2019
We study the problem of optimal transport on phylogenetic tree space from the perspective of tropical geometry, and thus define the Wasserstein-$p$ distances for probability measures in thisExpand
Geometry of Linear Convolutional Networks
• Computer Science, Mathematics
• ArXiv
• 2021
The theory predicts that the optimized parameters of an LCN will often correspond to repeated filters across layers, or filters that can be decomposed as repeated filters in the resulting function space. Expand
An Invitation to Tropical Alexandrov Curvature
• Mathematics
• 2021
We study Alexandrov curvature in the tropical projective torus with respect to the tropical metric. Alexandrov curvature is a generalization of classical Riemannian sectional curvature to moreExpand
Nonlinear Algebra and Applications
We showcase applications of nonlinear algebra in the sciences and engineering. Our survey is organized into eight themes: polynomial optimization, partial differential equations, algebraicExpand
Wasserstein distance to independence models
• Mathematics, Computer Science
• J. Symb. Comput.
• 2021
This work examines the combinatorial structure and algebraic degrees in the general case of the Segre-Veronese variety, and seeks to minimize its Wasserstein distance to a fixed independence model. Expand
1-Wasserstein distance on the standard simplex
• Mathematics
• 2019
Wasserstein distances provide a metric on a space of probability measures. We consider the space $\Omega$ of all probability measures on the finite set $\chi = \{1, \dots ,n\}$ where $n$ is aExpand

#### References

SHOWING 1-10 OF 28 REFERENCES
Tropical and Ordinary Convexity Combined
• Mathematics
• 2008
A polytrope is a tropical polytope which at the same time is convex in the ordinary sense. A $d$-dimensional polytrope turns out to be a tropical simplex, that is, it is the tropical convex hull ofExpand
Sharp asymptotic and finite-sample rates of convergence of empirical measures in Wasserstein distance
• Mathematics
• Bernoulli
• 2019
The Wasserstein distance between two probability measures on a metric space is a measure of closeness with applications in statistics, probability, and machine learning. In this work, we consider theExpand
On parameter estimation with the Wasserstein distance
• Mathematics
• Information and Inference: A Journal of the IMA
• 2019
Statistical inference can be performed by minimizing, over the parameter space, the Wasserstein distance between model distributions and the empirical distribution of the data. We study asymptoticExpand
Sinkhorn Distances: Lightspeed Computation of Optimal Transport
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. Expand
Dualities in Convex Algebraic Geometry
• Mathematics
• 2010
Convex algebraic geometry concerns the interplay between optimization theory and real algebraic geometry. Its objects of study include convex semialgebraic sets that arise in semidefinite programmingExpand
Discrete statistical models with rational maximum likelihood estimator
• Mathematics
• 2019
A discrete statistical model is a subset of a probability simplex. Its maximum likelihood estimator (MLE) is a retraction from that simplex onto the model. We characterize all models for which thisExpand
Mixtures and products in two graphical models
• Mathematics
• 2017
We compare two statistical models of three binary random variables. One is a mixture model and the other is a product of mixtures model called a restricted Boltzmann machine. Although the two modelsExpand
An Introduction to Polynomial and Semi-Algebraic Optimization
This is the first comprehensive introduction to the powerful moment approach for solving global optimization problems (and some related problems) described by polynomials (and even semi-algebraicExpand
Computational Optimal Transport
• Mathematics, Computer Science
• Found. Trends Mach. Learn.
• 2019
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. Expand
Algebraic Degree of Polynomial Optimization
• Mathematics, Computer Science
• SIAM J. Optim.
• 2009
It is proved that the optimality conditions always hold on optimizers, and the coordinates of optimizers are algebraic functions of the coefficients of the input polynomials, and a general formula is given for the algebraic degree of the optimal coordinates. Expand