Sample and Map from a Single Convex Potential: Generation using Conjugate Moment Measures

par Nina Vesseron (ENSAE)

mercredi 3 sept. 2025, 10:00 → 11:30 Europe/Paris

Current generative models typically follow a two-step process : first, they sample noise from a simple distribution, such as a Gaussian, and then learn a transformation to map this noise to the data distribution. In this talk, we explore a different approach that ties sampling and mapping together.

Our approach is inspired by the theory of moment measures [Cordero-Erausquin--Klartag 2015, Santambrogio 2016], which ensures that for any probability measure ρ, there exists a convex potential u such that ρ = ∇u ♯ exp(−u).

This decomposition naturally links the sampling step (from the log-concave distribution exp(−u)) with the mapping step (through ∇u). However, as we will see in simple examples—such as Gaussians or
one-dimensional distributions—this factorization is not suitable for practical generative tasks. For this reason, we introduce an alternative factorization in which ρ is decomposed as ρ = ∇w^*♯ exp(−w), where w^*
is the convex conjugate of w. We call this approach conjugate moment measures and show that it produces more intuitive results on these examples. Because ∇w^* is the Monge map between the log-concave distribution exp(−w) and ρ, we use neural optimal transport solvers to recover the convex potential w from samples of ρ.