Likelihood Matching for Diffusion Models

2025-08-05

AI Summary

This paper addresses the challenge of accurately modeling the reverse sampling path likelihood in diffusion models. To this end, we propose a likelihood matching training framework that establishes, for the first time, a rigorous equivalence between the target data distribution and the likelihood of the reverse diffusion path. By employing a quasi-likelihood approximation, we parameterize the reverse transition density as Gaussian and jointly optimize the score function and the Hessian matrix to match both the first-order (conditional mean) and second-order (conditional covariance) moments. Theoretically, we provide non-asymptotic convergence guarantees, prove the consistency of the quasi-maximum likelihood estimator, and quantify key factors influencing estimation error. Empirically, our method achieves superior generative quality and improved training stability compared to existing baselines.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Computer Vision: Diffusion Models for Vision
• Search and Optimization: Learning to Search

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• User Modeling, Personalization and Recommendation: Large Language Models (LLM) for user modeling and recommendation
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions

Abstract

We propose a Likelihood Matching approach for training diffusion models by first establishing an equivalence between the likelihood of the target data distribution and a likelihood along the sample path of the reverse diffusion. To efficiently compute the reverse sample likelihood, a quasi-likelihood is considered to approximate each reverse transition density by a Gaussian distribution with matched conditional mean and covariance, respectively. The score and Hessian functions for the diffusion generation are estimated by maximizing the quasi-likelihood, ensuring a consistent matching of both the first two transitional moments between every two time points. A stochastic sampler is introduced to facilitate computation that leverages on both the estimated score and Hessian information. We establish consistency of the quasi-maximum likelihood estimation, and provide non-asymptotic convergence guarantees for the proposed sampler, quantifying the rates of the approximation errors due to the score and Hessian estimation, dimensionality, and the number of diffusion steps. Empirical and simulation evaluations demonstrate the effectiveness of the proposed Likelihood Matching and validate the theoretical results.

Problem

• Establishes equivalence between target data likelihood and reverse diffusion path likelihood
• Estimates score and Hessian functions via Gaussian-approximated quasi-likelihood maximization
• Provides convergence guarantees for stochastic sampler addressing approximation errors

Innovation

• Likelihood Matching for diffusion models training
• Quasi-likelihood approximates reverse transition density
• Stochastic sampler uses score and Hessian information