Amortized Optimal Transport from Sliced Potentials

arXiv stat.ML / 4/17/2026

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Key Points

The paper proposes an amortized optimization framework to predict optimal transport (OT) plans across many pairs of measures using Kantorovich potentials obtained from sliced OT.
It introduces two training/inference strategies—regression-based amortization (RA-OT) and objective-based amortization (OA-OT)—to estimate functional models from slicedOT-derived potentials and recover OT plans from the estimated dual potentials.
Both approaches aim to make repeated OT computations much more efficient by reusing information learned from previous instances to rapidly approximate solutions for new measure pairs.
By leveraging sliced OT structure, the method is designed to be more parsimonious and less sensitive to representation details (e.g., the number of atoms in discrete settings) while maintaining high accuracy.
Experiments include MNIST digit transport, color transfer, supply-demand transport on spherical data, and mini-batch OT conditional flow matching to demonstrate practical effectiveness.

Abstract

We propose a novel amortized optimization method for predicting optimal transport (OT) plans across multiple pairs of measures by leveraging Kantorovich potentials derived from sliced OT. We introduce two amortization strategies: regression-based amortization (RA-OT) and objective-based amortization (OA-OT). In RA-OT, we formulate a functional regression model that treats Kantorovich potentials from the original OT problem as responses and those obtained from sliced OT as predictors, and estimate these models via least-squares methods. In OA-OT, we estimate the parameters of the functional model by optimizing the Kantorovich dual objective. In both approaches, the predicted OT plan is subsequently recovered from the estimated potentials. As amortized OT methods, both RA-OT and OA-OT enable efficient solutions to repeated OT problems across different measure pairs by reusing information learned from prior instances to rapidly approximate new solutions. Moreover, by exploiting the structure provided by sliced OT, the proposed models are more parsimonious, independent of specific structures of the measures, such as the number of atoms in the discrete case, while achieving high accuracy. We demonstrate the effectiveness of our approaches on tasks including MNIST digit transport, color transfer, supply-demand transportation on spherical data, and mini-batch OT conditional flow matching.