MGD: Moment Guided Diffusion for Maximum Entropy Generation

TL;DR

The paper introduces Moment Guided Diffusion (MGD), a novel method combining diffusion models and maximum entropy principles to efficiently generate high-dimensional data with controlled moments, avoiding slow mixing issues.

Contribution

MGD is a new approach that guides diffusion processes to produce maximum entropy distributions with finite-time moment control, providing theoretical guarantees and practical estimators.

Findings

MGD converges to maximum entropy distribution in large-volatility limit.

The method provides a tractable entropy estimator from dynamics.

Applied to complex data, MGD estimates negentropy in high-dimensional processes.

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.