Brenier’s theorem links optimal transport, statistics, and diffusion models
docmilanfar · x · 2026-07-22
Brenier’s theorem connects optimal transport with statistics and machine learning.
- The thread explains that the theorem gives a unique optimal map between two densities under an $L2$ cost, with the map written as the gradient of a convex potential.
- It also highlights a second formulation: a vector field can be decomposed into a convex-gradient part and a measure-preserving rearrangement.
- The author argues this result underpins ideas that later showed up in diffusion models, especially potential flows and probability flows.
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