Energy Generative Modeling: A Lyapunov-based Energy Matching Perspective

2026-05-06

AI Summary

This work addresses the disconnection between training and sampling in static scalar energy-based generative models, the absence of a unified theoretical framework, and the lack of convergence guarantees for deterministic gradient flows. The authors unify these aspects by formulating the problem as density transport in Wasserstein space, constructing a nonlinear control system with the KL divergence serving as a Lyapunov function, where training and sampling differ only in their initial conditions. Key contributions include the first Lyapunov-stability-theory-based unified generative framework, a stopping criterion for finite-step Langevin sampling, and a proof that energy superposition preserves the Gibbs invariant measure while inheriting the Lyapunov certificate. Experiments corroborate theoretical predictions, validating the stopping criterion, the invariant measure property, and the failure of deterministic gradient flows to satisfy Lyapunov convergence conditions.

Abstract

Generative models based on static scalar energy functions represent an emerging paradigm in which a single time independent potential drives sample generation through its gradient field, eliminating the need for time conditioning entirely. We unify the training and sampling phases of this paradigm, conventionally treated as separate procedures, within a single framework: density transport on the Wasserstein space, cast as a nonlinear control problem in which the Kullback Leibler (KL) divergence serves as a Lyapunov function. Training and sampling are then two instances of this same master dynamics, differing only in initial condition. Within this autonomous framework we develop two analytic results. First, since the Lyapunov certificate is asymptotic, we derive a finite step stopping criterion for Langevin sampling and prove that no Lyapunov certificate exists for the deterministic gradient flow on the same energy landscape. Second, the reformulation brings the toolkit of nonlinear control theory to bear on static scalar energy generative modeling, that is, we show that additive composition of trained scalar energies retains an explicit Gibbs invariant measure and inherits the closed-loop Lyapunov certificate. Beyond these immediate results, this reformulation bridges static scalar energy generative models with the full toolkit of nonlinear control theory, opening the door to barrier functions for constrained generation and contraction metrics for accelerated sampling. Experiments on synthetic distributions validate the theoretical predictions.

Problem

• Energy Generative Modeling
• Lyapunov Function
• Wasserstein Space
• Nonlinear Control
• KL Divergence

Innovation

• Energy-based generative modeling
• Lyapunov stability
• Wasserstein gradient flow
• Nonlinear control theory
• KL divergence