MGD: Moment Guided Diffusion for Maximum Entropy Generation
Abstract
Generating samples from limited information is a fundamental problem across scientific domains. Classical maximum entropy methods provide principled uncertainty quantification from moment constraints but require sampling via MCMC or Langevin dynamics, which typically exhibit exponential slowdown in high dimensions. In contrast, generative models based on diffusion and flow matching efficiently transport noise to data but offer limited theoretical guarantees and can overfit when data is scarce. We introduce Moment Guided Diffusion (MGD), which combines elements of both approaches. Building on the stochastic interpolant framework, MGD samples maximum entropy distributions by solving a stochastic differential equation that guides moments toward prescribed values in finite time, thereby avoiding slow mixing in equilibrium-based methods. We formally obtain, in the large-volatility limit, convergence of MGD to the maximum entropy distribution and derive a tractable estimator of the resulting entropy computed directly from the dynamics. Applications to financial time series, turbulent flows, and cosmological fields using wavelet scattering moments yield estimates of negentropy for high-dimensional multiscale processes.
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BibTeX
@misc{lempereur2026moment,
title = {{MGD: Moment Guided Diffusion for Maximum Entropy Generation}},
author = {Etienne Lempereur and Nathanaël Cuvelle--Magar and Florentin Coeurdoux and Stéphane Mallat and Eric Vanden-Eijnden},
howpublished = {HAL},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2602.17211},
eprint = {2602.17211},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05843010v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - MGD: Moment Guided Diffusion for Maximum Entropy Generation
AU - Etienne Lempereur
AU - Nathanaël Cuvelle--Magar
AU - Florentin Coeurdoux
AU - Stéphane Mallat
AU - Eric Vanden-Eijnden
PY - 2026
DA - 2026/10/28
PB - HAL
DO - 10.48550/arxiv.2602.17211
UR - https://hal.science/hal-05843010v1
AB - Generating samples from limited information is a fundamental problem across scientific domains. Classical maximum entropy methods provide principled uncertainty quantification from moment constraints but require sampling via MCMC or Langevin dynamics, which typically exhibit exponential slowdown in high dimensions. In contrast, generative models based on diffusion and flow matching efficiently transport noise to data but offer limited theoretical guarantees and can overfit when data is scarce. We introduce Moment Guided Diffusion (MGD), which combines elements of both approaches. Building on the stochastic interpolant framework, MGD samples maximum entropy distributions by solving a stochastic differential equation that guides moments toward prescribed values in finite time, thereby avoiding slow mixing in equilibrium-based methods. We formally obtain, in the large-volatility limit, convergence of MGD to the maximum entropy distribution and derive a tractable estimator of the resulting entropy computed directly from the dynamics. Applications to financial time series, turbulent flows, and cosmological fields using wavelet scattering moments yield estimates of negentropy for high-dimensional multiscale processes.
ER -