On the Equivalence of Optimal Transport Problem and Action Matching with Optimal Vector Fields

TL;DR

This paper demonstrates that focusing solely on optimal vector fields in flow matching and action matching methods can effectively solve optimal transport problems, offering a new perspective on distribution mapping in generative modeling.

Contribution

It establishes the equivalence between optimal transport and action matching by showing that optimal vector fields suffice for OT in both frameworks.

Findings

Optimal vector fields are sufficient for solving OT problems.

Action Matching can be used to achieve OT without manual interpolation.

The approach unifies flow matching and action matching under OT theory.

Abstract

Flow Matching (FM) method in generative modeling maps arbitrary probability distributions by constructing an interpolation between them and then learning the vector field that defines ODE for this interpolation. Recently, it was shown that FM can be modified to map distributions optimally in terms of the quadratic cost function for any initial interpolation. To achieve this, only specific optimal vector fields, which are typical for solutions of Optimal Transport (OT) problems, need to be considered during FM loss minimization. In this note, we show that considering only optimal vector fields can lead to OT in another approach: Action Matching (AM). Unlike FM, which learns a vector field for a manually chosen interpolation between given distributions, AM learns the vector field that defines ODE for an entire given sequence of distributions.