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

The canonical approach in generative modeling is to split model fitting into two blocks: define first how to sample noise (e.g. Gaussian) and choose next what to do with it (e.g. using a single map or flows). We explore in this work an alternative route that ties sampling and mapping. We find inspiration in moment measures, a result that states that for any measure ρ, there exists a unique convex potential u such that ρ = ∇u♯e-u. While this does seem to tie effectively sampling (from log–concave distribution e-u) and action (pushing particles through ∇u), we observe on simple examples (e.g…