SIAM Journal on Scientific Computing· 2015Q1
Iterative Bregman Projections for Regularized Transportation Problems
- 689citations
- Q1SCImago
- 2015year
Short summary
A new numerical framework uses entropic regularization and iterative Bregman projections to efficiently approximate solutions for optimal transport problems, particularly those with complex constraints.
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Key points
- Introduces entropic regularization for optimal transport problems.
- Uses iterative Bregman projections for efficient constraint satisfaction.
- Demonstrates applicability to diverse problems including barycenters and fluid dynamics.
- Handles both equality and inequality constraints via Bregman--Dykstra iterations.
AI-generated from the title and abstract; the full text is not read.
Abstract
This paper details a general numerical framework to approximate solutions to linear programs related to optimal transport. The general idea is to introduce an entropic regularization of the initial linear program. This regularized problem corresponds to a Kullback--Leibler Bregman divergence projection of a vector (representing some initial joint distribution) on the polytope of constraints. We show 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. This allows us to make use of iterative Bregman projections (when there are only equality constraints) or, more generally, Bregman--Dykstra iterations (when inequality constraints are involved). We illustrate the usefulness of this approach for several variational problems related to optimal transport: barycenters for the optimal transport metric, tomographic reconstruction, multimarginal optimal transport, and in particular its application to Brenier's relaxed solutions of incompressible Euler equations, partial unbalanced optimal transport, and optimal transport with capacity constraints.
The authors' abstract, as published at the source. SIAM Journal on Scientific Computing, 2015 · DOI ↗
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Field: Applied Mathematics
Applied MathematicsMathematics