[Paper Review] Iterative Bregman Projections for Regularized Transportation Problems

Jean‐David Benamou, Guillaume Carlier+3 more|arXiv (Cornell University)|Dec 16, 2014

Risk and Portfolio Optimization46 references3 citations

TL;DR

This paper introduces a unified numerical framework based on iterative Bregman projections to efficiently solve regularized optimal transport problems by leveraging entropic regularization. By reformulating the problem as a Kullback-Leibler divergence projection onto a polytope of constraints, the method enables fast, scalable solutions through closed-form projections on simple constraint sets, achieving high accuracy with low computational cost across diverse applications including barycenters, tomography, and multi-marginal transport.

ABSTRACT

This article details a general numerical framework to approximate so-lutions 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 di-vergence projection of a vector (representing some initial joint distribu-tion) 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 in-volved). We illustrate the usefulness of this approach to several variational problems related to optimal transport: barycenters for the optimal trans-port metric, tomographic reconstruction, multi-marginal optimal trans-port and in particular its application to Brenier's relaxed solutions of in-compressible Euler equations, partial un-balanced optimal transport and optimal transport with capacity constraints.

Motivation & Objective

• To develop a general, efficient numerical method for solving linear programs arising in optimal transport and its variants.
• To address the computational bottleneck of exact optimal transport by introducing a smooth, regularized formulation.
• To unify the solution of diverse optimal transport problems—such as barycenters, partial transport, and multi-marginal transport—under a single algorithmic framework.
• To enable scalable computation on high-dimensional problems (e.g., 2D grids) that are intractable with standard linear programming.
• To demonstrate the effectiveness of iterative KL projections via Dykstra and Bregman methods on complex transport constraints.

Proposed method

• The method introduces entropic regularization to transform the original optimal transport problem into a strictly convex, smooth optimization problem.
• The regularized problem is recast as a Kullback-Leibler (KL) Bregman projection of an initial joint distribution onto the polytope defined by marginal and capacity constraints.
• For problems with multiple constraints, the algorithm uses iterative Bregman or Dykstra's projections, exploiting closed-form solutions for individual constraint sets.
• Projections onto marginal constraints are computed via scaling operations: for each marginal, the solution is adjusted by the ratio of target to current mass, capped at 1.
• The projection onto total mass constraints is a simple normalization of the current iterate by the total mass.
• The algorithm alternates between projections on each constraint set until convergence, leveraging the structure of the entropic regularizer and KL divergence.

Experimental results

Research questions

• RQ1 Can iterative Bregman projections provide an efficient and scalable solution to regularized optimal transport problems?
• RQ2 How can entropic regularization be leveraged to enable closed-form projections on complex constraint sets in optimal transport?
• RQ3 Can this framework unify the solution of diverse optimal transport variants, such as barycenters, partial transport, and multi-marginal problems?
• RQ4 What is the computational and numerical performance of this method on high-dimensional problems like 2D grids?
• RQ5 How does the method preserve structural symmetries (e.g., self-duality) in solutions compared to exact linear programming?

Key findings

• The method achieves high accuracy with low computational cost, enabling solutions on 2D grids of size 50×50 and 60×60, which are computationally prohibitive for standard linear programming.
• For 1D and 2D test cases, the computed transport plans preserve expected symmetries (e.g., self-duality and rotational invariance), confirming correctness of the solution.
• The algorithm converges reliably for problems involving multi-marginal transport with partial mass constraints, as demonstrated on three-marginal problems with m=0.7×min(⟨pk,1⟩).
• The solution to the regularized problem is a diagonal scaling of e^{-C/ε}, which allows efficient computation via the iterative proportional fitting procedure (IPFP).
• The framework successfully handles complex constraints such as capacity limits and partial transport, with projections computed in closed form for each constraint set.
• The method outperforms exact linear programming in scalability and is well-suited for applications in machine learning, image processing, and computational economics.