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Fast Transport Optimization for Monge Costs on the Circle

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Date
2010
Link to item file
http://hal.archives-ouvertes.fr/hal-00362834/en/
Dewey
Probabilités et mathématiques appliquées
Sujet
Monge-Kantorovich problem; Aubry-Mather (weak KAM) theory; Optimization and Control
Journal issue
SIAM Journal on Applied Mathematics
Volume
70
Number
7
Article pages
2239-2258
Publisher
SIAM
DOI
http://dx.doi.org/10.1137/090772708
URI
https://basepub.dauphine.fr/handle/123456789/2954
Collections
  • CEREMADE : Publications
Metadata
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Author
Delon, Julie
Salomon, Julien
Sobolevski, Andrei
Type
Article accepté pour publication ou publié
Abstract (EN)
Consider the problem of optimally matching two measures on the circle, or equivalently two periodic measures on the real line, and suppose the cost of matching two points satisfies the Monge condition. We introduce a notion of locally optimal transport plan, motivated by the weak KAM (Aubry-Mather) theory, and show that all locally optimal transport plans are conjugate to shifts. This theory is applied to a transportation problem arising in image processing: for two sets of point masses, both of which have the same total mass, find an optimal transport plan with respect to a given cost function that satisfies the Monge condition. For the case of N real-valued point masses we present an O(N log epsilon) algorithm that approximates the optimal cost within epsilon; when all masses are integer multiples of 1/M, the algorithm gives an exact solution in O(N log M) operations.

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