A Push-Relabel Based Additive Approximation for Optimal Transport

by   Nathaniel Lahn, et al.

Optimal Transport is a popular distance metric for measuring similarity between distributions. Exact algorithms for computing Optimal Transport can be slow, which has motivated the development of approximate numerical solvers (e.g. Sinkhorn method). We introduce a new and very simple combinatorial approach to find an ε-approximation of the OT distance. Our algorithm achieves a near-optimal execution time of O(n^2/ε^2) for computing OT distance and, for the special case of the assignment problem, the execution time improves to O(n^2/ε). Our algorithm is based on the push-relabel framework for min-cost flow problems. Unlike the other combinatorial approach (Lahn, Mulchandani and Raghvendra, NeurIPS 2019) which does not have a fast parallel implementation, our algorithm has a parallel execution time of O(log n/ε^2). Interestingly, unlike the Sinkhorn algorithm, our method also readily provides a compact transport plan as well as a solution to an approximate version of the dual formulation of the OT problem, both of which have numerous applications in Machine Learning. For the assignment problem, we provide both a CPU implementation as well as an implementation that exploits GPU parallelism. Experiments suggest that our algorithm is faster than the Sinkhorn algorithm, both in terms of CPU and GPU implementations, especially while computing matchings with a high accuracy.


page 1

page 2

page 3

page 4


A Graph Theoretic Additive Approximation of Optimal Transport

Transportation cost is an attractive similarity measure between probabil...

Computing the L1 optimal transport density: a FEM approach

The L^1 optimal transport density μ^* is the unique L^∞ solution of the ...

On the Convergence of Gradient Extrapolation Methods for Unbalanced Optimal Transport

We study the Unbalanced Optimal Transport (UOT) between two measures of ...

Fast Algorithms for a New Relaxation of Optimal Transport

We introduce a new class of objectives for optimal transport computation...

Dynamic multi-agent assignment via discrete optimal transport

We propose an optimal solution to a deterministic dynamic assignment pro...

Combining Reinforcement Learning and Optimal Transport for the Traveling Salesman Problem

The traveling salesman problem is a fundamental combinatorial optimizati...

Tuning symplectic integrators is easy and worthwhile

Many applications in computational physics that use numerical integrator...

Please sign up or login with your details

Forgot password? Click here to reset