Amortized Optimal Transport from Sliced Potentials

Researchers Minh-Phuc Truong and Khai Nguyen have introduced a groundbreaking approach to optimal transport (OT) problems with their paper 'Amortized Optimal Transport from Sliced Potentials.' The core innovation lies in using Kantorovich potentials derived from sliced OT—a computationally simpler version of OT—to predict solutions for more complex transport problems. By amortizing the learning process across multiple problem instances, their method dramatically reduces the computational burden of solving repeated OT problems, which traditionally require expensive optimization for each new pair of probability distributions.

The researchers propose two distinct amortization strategies: regression-based amortization (RA-OT) and objective-based amortization (OA-OT). RA-OT treats potentials from the original OT problem as responses to be predicted from sliced OT potentials using functional regression models, while OA-OT directly optimizes the Kantorovich dual objective. Both approaches enable rapid approximation of new solutions by reusing information from previously solved instances, achieving high accuracy while being independent of specific data structures like the number of atoms in discrete cases.

In practical demonstrations, the method showed remarkable effectiveness across diverse applications including MNIST digit transport, color transfer between images, supply-demand transportation on spherical data, and mini-batch OT conditional flow matching for generative models. The approach's parsimonious structure and computational efficiency make it particularly valuable for machine learning applications where OT calculations are repeatedly needed, such as in training generative models or solving matching problems in computer vision.