Sample and Map from a Single Convex Potential: Generation using Conjugate Moment Measures
arXiv:2503.10576v1 Announce Type: new
Abstract: A common approach to 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 $rho$ supported on a compact convex set of $mathbb{R}^d$, there exists a unique convex potential $u$ such that $rho=nabla u,sharp,e^{-u}$. 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 $rho$ is factorized as $nabla w^*,sharp,e^{-w}$, where $w^*$ is the convex conjugate of $w$. 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 $rho$, we rely on optimal transport solvers to propose an algorithm to recover $w$ from samples of $rho$, and parameterize $w$ as an input-convex neural network.
Abstract: A common approach to 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 $rho$ supported on a compact convex set of $mathbb{R}^d$, there exists a unique convex potential $u$ such that $rho=nabla u,sharp,e^{-u}$. 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 $rho$ is factorized as $nabla w^*,sharp,e^{-w}$, where $w^*$ is the convex conjugate of $w$. 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 $rho$, we rely on optimal transport solvers to propose an algorithm to recover $w$ from samples of $rho$, and parameterize $w$ as an input-convex neural network.
Nina Vesseron, Louis B’ethune, Marco Cuturi
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