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

2025-03-13

AI Summary

This work addresses the modeling inefficiency in generative modeling arising from the decoupling of noise sampling and transport mapping. We propose the Conjugate Moment Measure (CMM) framework, which represents the target distribution ρ as the pushforward of a log-concave base distribution e⁻ʷ under ∇w*, where w* is the Legendre conjugate of a convex function w. CMM intrinsically unifies sampling and mapping—resolving the ill-posedness of conventional moment measures in generation—and establishes theoretical equivalence between ∇w* and the optimal Monge map in optimal transport. We parameterize w using input-convex neural networks (ICNNs) and construct the generator via dual optimization and gradient mapping. Experiments on Gaussian and 1D benchmarks demonstrate substantial improvements in sample quality and geometric fidelity. The framework introduces a novel paradigm for latent-space modeling grounded in convex analysis and optimal transport theory.

Technology Category

• Machine Learning: Deep Generative Models & Autoencoders
• Computer Vision: Generative Adversarial Networks (GANs) for Vision
• Search and Optimization: Non-convex Optimization

Application Category

• Web Mining and Content Analysis: Web data generation and simulation
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs

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 $ ho$ supported on a compact convex set of $mathbb{R}^d$, there exists a unique convex potential $u$ such that $ ho= abla u,sharp,e^{-u}$. While this does seem to tie effectively sampling (from log-concave distribution $e^{-u}$) and action (pushing particles through $ abla 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 $ ho$ is factorized as $ abla 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 $ abla w^*$ is the Monge map between the log-concave distribution $e^{-w}$ and $ ho$, we rely on optimal transport solvers to propose an algorithm to recover $w$ from samples of $ ho$, and parameterize $w$ as an input-convex neural network.

Problem

• Explores tying sampling and mapping in generative modeling.
• Proposes conjugate moment measures for more intuitive results.
• Uses optimal transport to recover parameters from samples.

Innovation

• Combines sampling and mapping via conjugate moment measures.
• Uses optimal transport solvers to recover potential functions.
• Parameterizes convex potentials with input-convex neural networks.