Regularity of the score function in generative models

2025/06/24 by Arthur Stéphanovitch, Stéphanovitch, Arthur · 4 citations
Physics and Astronomy · Psychology · #Convergence (economics) #FOS: Mathematics #Function (biology) #Generative grammar #Generative model #Lipschitz continuity #Mental Health Research Topics #Opinion Dynamics and Social Influence #Smoothness #Stability (learning theory) #Statistics Theory (math.ST)

paper · pdf · doi:10.48550/arxiv.2506.19559

published in arXiv (Cornell University) (Cornell University)

openalex publication_date 2025/06/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

We study the regularity of the score function in score-based generative models and show that it naturally adapts to the smoothness of the data distribution. Under minimal assumptions, we establish Lipschitz estimates that directly support convergence and stability analyses in both diffusion and ODE-based generative models. In addition, we derive higher-order regularity bounds, which simplify existing arguments for optimally approximating the score function using neural networks.

Citations

• A coupling approach to Lipschitz transport maps
• Beyond Log-Concavity and Score Regularity: Improved Convergence Bounds for Score-Based Generative Models in W2-distance
• Smooth transport map via diffusion process
• Flow matching achieves almost minimax optimal convergence
• Heat flow, log-concavity, and Lipschitz transport maps
• Transport equations and flows with one-sided Lipschitz velocity fields
• Error Bounds for Flow Matching Methods
• Improved Convergence of Score-Based Diffusion Models via Prediction-Correction
• The probability flow ODE is provably fast
• Transportation onto log-Lipschitz perturbations
• Diffusion Models are Minimax Optimal Distribution Estimators
• Improved Analysis of Score-based Generative Modeling: User-Friendly Bounds under Minimal Smoothness Assumptions
• Convergence of score-based generative modeling for general data distributions
• Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions
• Elucidating the Design Space of Diffusion-Based Generative Models
• Equivariant Diffusion for Molecule Generation in 3D
• On the Lipschitz Properties of Transportation Along Heat Flows
• Diffusion Models Beat GANs on Image Synthesis
• Score-Based Generative Modeling through Stochastic Differential Equations
• DiffWave: A Versatile Diffusion Model for Audio Synthesis
• Adaptivity of deep ReLU network for learning in Besov and mixed smooth Besov spaces: optimal rate and curse of dimensionality
• A Generalization of Caffarelli's Contraction Theorem via (reverse) Heat Flow

Cited by

• Wasserstein Convergence of Critically Damped Langevin Diffusions
• Mimicking diffusion processes with differential equations
• Distribution estimation via Flow Matching with Lipschitz guarantees
• Generalization bounds for score-based generative models: a synthetic proof

Related

• Generalization bounds for score-based generative models: a synthetic proof
• Improved Analysis of Score-based Generative Modeling: User-Friendly Bounds under Minimal Smoothness Assumptions
• Global Well-posedness and Convergence Analysis of Score-based Generative Models via Sharp Lipschitz Estimates
• Convergence of score-based generative modeling for general data distributions
• Algorithm- and Data-Dependent Generalization Bounds for Diffusion Models