Convergence Analysis of Probability Flow ODE for Score-based Generative Models
This work addresses a theoretical gap for researchers in generative modeling, providing rigorous convergence analysis that is incremental but essential for advancing the field.
The paper tackles the problem of understanding the convergence properties of deterministic samplers based on probability flow ODEs in score-based generative models, proving theoretical error bounds in terms of data dimension and score matching error, and validating these with numerical studies up to 128 dimensions.
Score-based generative models have emerged as a powerful approach for sampling high-dimensional probability distributions. Despite their effectiveness, their theoretical underpinnings remain relatively underdeveloped. In this work, we study the convergence properties of deterministic samplers based on probability flow ODEs from both theoretical and numerical perspectives. Assuming access to $L^2$-accurate estimates of the score function, we prove the total variation between the target and the generated data distributions can be bounded above by $\mathcal{O}(d^{3/4}δ^{1/2})$ in the continuous time level, where $d$ denotes the data dimension and $δ$ represents the $L^2$-score matching error. For practical implementations using a $p$-th order Runge-Kutta integrator with step size $h$, we establish error bounds of $\mathcal{O}(d^{3/4}δ^{1/2} + d\cdot(dh)^p)$ at the discrete level. Finally, we present numerical studies on problems up to 128 dimensions to verify our theory.