Research map / Probabilistic and scientific modeling / Variational and generative sampling
Beyond Log-Concavity and Score Regularity: Improved Convergence Bounds for Score-Based Generative Models in W2-distance
Read the paperCheck your idea against 108,967 papers
AI-extracted summary. Written by AI from the paper's full text. Numbers and claims can be wrong or missing context, so check the paper before you cite or build on it.
Gives W2-convergence bounds for OU-based score-based generative models under weak log-concavity and one-sided log-Lipschitz assumptions on the data, with no regularity assumed on the score or its estimator, via a PDE (HJB) and coupling analysis.
Problem
Existing W2 convergence bounds for score-based generative models rely on strong assumptions such as log-concavity of the data distribution and regularity of the score function or its estimator.
Approach
Shows that weak log-concavity of the data propagates along the OU forward flow into log-concavity, uses HJB/PDE analysis and a stochastic-control-inspired coupling to bound the discretization, initialization and score-estimation errors in W2 using only weak convexity and one-sided Lipschitz conditions on the data.
Key results
- Main theorem: W2(p_data, L(X*_tN)) <= C(e^{-T} W2(p_data, pi_inf) + eps T + sqrt(h)(sqrt(d)+sqrt(m2)) T), with C exponential in alpha, M, L_U.
- Weak log-concavity constant evolves so that the forward marginals become strongly log-concave after time xi(alpha,M) = log(sqrt((alpha^2+M-alpha)/alpha^2)) ∧ T when alpha-M<0.
- Gaussian mixtures with equal variance are shown to be weakly convex with alpha=1/sigma^2 and Lipschitz score.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use weak log-concavity and one-sided log-Lipschitz assumptions instead of strong log-concavity and strict score regularity | OU-based score-based generative models · W2 convergence | W2 convergence guarantee | Helped |
| use the forward OU process to regularize weakly log-concave data | OU-based score-based generative models · Forward OU flow | log-concavity | Helped |
| apply OU Gaussian-kernel regularization to Gaussian mixtures | Gaussian mixture distributions · Early stopping | stability | Helped |
Surprising findings
- The backward OU drift alternates between non-contractive and contractive regimes, with a transition time T(alpha,M,0) before which contractivity is not guaranteed.
Limitations
- Bound depends exponentially on alpha, M and L_U.
- Analysis is theoretical with no experiments; Gaussian mixture example only for constants.
Experimental setup
Metrics: W2 distance bound
Baselines: KL-divergence convergence bounds for SGMs W2 bounds under log-concavity (proximal sampler / strongly log-concave)
Builds on: Ornstein-Uhlenbeck forward process Euler-Maruyama discretization coupling by reflection once log-concave always log-concave
Closest papers by meaning
- Global Well-posedness and Convergence Analysis of Score-based Generative Models via Sharp Lipschitz Estimates (ICLR 2025 · similarity 0.90)
- Algorithm- and Data-Dependent Generalization Bounds for Diffusion Models (NeurIPS 2025 · similarity 0.89)
- Score-based generative models are provably robust: an uncertainty quantification perspective (NeurIPS 2024 · similarity 0.89)
- Sampling is as easy as keeping the consistency: convergence guarantee for Consistency Models (ICML 2024 · similarity 0.88)
- Dimension-free Score Matching and Time Bootstrapping for Diffusion Models (NeurIPS 2025 · similarity 0.87)
- A Non-Asymptotic Convergent Analysis for Scored-Based Graph Generative Model via a System of Stochastic Differential Equations (ICML 2025 · similarity 0.87)
Writing a paper in this area? scime draft is a free LaTeX editor with AI included, real-time co-authors and a check of your references and venue rules before you submit. It also imports OpenAI Prism projects.