Sample and Map from a Single Convex Potential: Generation using Conjugate Moment Measures
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| Format: | Preprint |
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2025
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| _version_ | 1866913859870130176 |
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| author | Vesseron, Nina Béthune, Louis Cuturi, Marco |
| author_facet | Vesseron, Nina Béthune, Louis Cuturi, Marco |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_10576 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sample and Map from a Single Convex Potential: Generation using Conjugate Moment Measures Vesseron, Nina Béthune, Louis Cuturi, Marco Machine Learning 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. |
| title | Sample and Map from a Single Convex Potential: Generation using Conjugate Moment Measures |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2503.10576 |