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

arXiv: https://arxiv.org/abs/2503.10576 · HTML: https://mlsanity.com/paper/2503.10576/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 41 references that are in the map.
- First posted on arXiv 13 Mar 2025.

## Builds on (most cited first, 22 of 22)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Building Normalizing Flows with Stochastic Interpolants](https://mlsanity.com/paper/2209.15571/) (2022)
- [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023)
- [FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Models](https://mlsanity.com/paper/1810.01367/) (2018)
- [Input Convex Neural Networks](https://mlsanity.com/paper/1609.07152/) (2016)
- [Multisample Flow Matching: Straightening Flows with Minibatch Couplings](https://mlsanity.com/paper/2304.14772/) (2023)
- [VAE with a VampPrior](https://mlsanity.com/paper/1705.07120/) (2017)
- [Theoretical guarantees for approximate sampling from smooth and log-concave densities](https://mlsanity.com/paper/1412.7392/) (2014)
- [Optimal transport mapping via input convex neural networks](https://mlsanity.com/paper/1908.10962/) (2019)
- [On Wasserstein Two Sample Testing and Related Families of Nonparametric Tests](https://mlsanity.com/paper/1509.02237/) (2015)
- [User-friendly guarantees for the Langevin Monte Carlo with inaccurate gradient](https://mlsanity.com/paper/1710.00095/) (2017)
- [Wasserstein-2 Generative Networks](https://mlsanity.com/paper/1909.13082/) (2019)
- [Do Neural Optimal Transport Solvers Work? A Continuous Wasserstein-2 Benchmark](https://mlsanity.com/paper/2106.01954/) (2021)
- [Convergence of Langevin MCMC in KL-divergence](https://mlsanity.com/paper/1705.09048/) (2017)
- [GMMSeg: Gaussian Mixture based Generative Semantic Segmentation Models](https://mlsanity.com/paper/2210.02025/) (2022)
- [On amortizing convex conjugates for optimal transport](https://mlsanity.com/paper/2210.12153/) (2022)
- [XGAN: Unsupervised Image-to-Image Translation for Many-to-Many Mappings](https://mlsanity.com/paper/1711.05139/) (2017)
- [On a Neural Implementation of Brenier's Polar Factorization](https://mlsanity.com/paper/2403.03071/) (2024)

## Built on by (most cited first, 2 of 2)

- [Estimation of Stochastic Optimal Transport Maps](https://mlsanity.com/paper/2512.09499/) (2025)
- [A lift for input-convex neural net training](https://mlsanity.com/paper/2605.24274/) (2026)

## Abstract (arXiv)

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=nabla u sharp e-u. While this does seem to tie effectively sampling (from log-concave distribution e-u) and action (pushing particles through nabla u), we observe on simple examples (e.g., Gaussians or 1D distributions) that this choice is ill-suited for practical tasks. We study an alternative factorization, where ρ is factorized as nabla w^*sharp e-w, where w^* is the convex conjugate of a convex potential w. We call this approach conjugate moment measures, and show far more intuitive results on these examples. Because nabla w^* is the Monge map between the log-concave distribution e-w and ρ, we rely on optimal transport solvers to propose an algorithm to recover w from samples of ρ, and parameterize w as an input-convex neural network. We also address the common sampling scenario in which the density of ρ is known only up to a normalizing constant, and propose an algorithm to learn w in this setting.