On the Equivalence of Optimal Transport Problem and Action Matching with Optimal Vector Fields
Mapping between sequences of continuous probability distributions arises in generative modeling, yet existing optimal transport (OT) methods suffer from high computational cost due to explicit joint distribution estimation or Wasserstein distance computation. Method: We establish a rigorous equivalence between Action Matching (AM) and OT by formulating distributional transport as learning an optimal vector field for a generative ordinary differential equation (ODE). Under compatible boundary conditions and energy functional design, the AM solution inherently satisfies the Kantorovich duality optimality criterion. Contribution/Results: This equivalence reveals that OT’s optimal coupling is fully characterized by a specific class of integrable vector fields—bypassing explicit joint distribution optimization or Wasserstein metric evaluation. Empirically, AM preserves theoretical optimality while achieving substantial computational efficiency gains. Our work provides a novel theoretical foundation and optimization paradigm for flow-matching-based generative modeling.