The Sinkhorn algorithm, parabolic optimal transport and geometric Monge–Ampère equations

We show that the discrete Sinkhorn algorithm—as applied in the setting of Optimal Transport on a compact manifold—converges to the solution of a fully non-linear parabolic PDE of Monge–Ampère type, in a large-scale limit. The latter evolution equation has previously appeared in different contexts (e...

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Bibliographic Details
Published in:Numerische Mathematik Vol. 145; no. 4; pp. 771 - 836
Main Author: Berman, Robert J.
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.08.2020
Springer Nature B.V
Subjects:
Algorithms
boundary-value problem
Complexity
convergence
Matematik
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematical sciences
Mathematics
Mathematics and Statistics
Monte Carlo simulation
neumann
Numerical Analysis
Numerical and Computational Physics
numerical-solution
Parabolic differential equations
Partial differential equations
polar factorization
regularity
schemes
Simulation
Theoretical
Toruses
35J60
65N99
35K55
90C08
ISSN:0029-599X, 0945-3245, 0945-3245
Online Access:Get full text
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Summary:We show that the discrete Sinkhorn algorithm—as applied in the setting of Optimal Transport on a compact manifold—converges to the solution of a fully non-linear parabolic PDE of Monge–Ampère type, in a large-scale limit. The latter evolution equation has previously appeared in different contexts (e.g. on the torus it can be be identified with the Ricci flow). This leads to algorithmic approximations of the potential of the Optimal Transport map, as well as the Optimal Transport distance, with explicit bounds on the arithmetic complexity of the construction and the approximation errors. As applications we obtain explicit schemes of nearly linear complexity, at each iteration, for optimal transport on the torus and the two-sphere, as well as the far-field antenna problem. Connections to Quasi-Monte Carlo methods are exploited.
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ISSN:0029-599X
0945-3245
0945-3245
DOI:10.1007/s00211-020-01127-x