Assessing the Quality of Denoising Diffusion Models in Wasserstein Distance: Noisy Score and Optimal Bounds

TL;DR

This paper evaluates the robustness and convergence of denoising diffusion models using Wasserstein distance, providing theoretical guarantees and empirical evidence of their optimality and resilience to noisy score estimates.

Contribution

It offers the first finite-sample guarantees for DDPMs in Wasserstein distance, demonstrating robustness to noise and faster convergence rates than prior results.

Findings

DDPMs are robust to constant-variance noise in score evaluations.

Finite-sample guarantees in Wasserstein-2 distance are established.

Convergence rates match Gaussian case, indicating optimality.

Abstract

Generative modeling aims to produce new random examples from an unknown target distribution, given access to a finite collection of examples. Among the leading approaches, denoising diffusion probabilistic models (DDPMs) construct such examples by mapping a Brownian motion via a diffusion process driven by an estimated score function. In this work, we first provide empirical evidence that DDPMs are robust to constant-variance noise in the score evaluations. We then establish finite-sample guarantees in Wasserstein-2 distance that exhibit two key features: (i) they characterize and quantify the robustness of DDPMs to noisy score estimates, and (ii) they achieve faster convergence rates than previously known results. Furthermore, we observe that the obtained rates match those known in the Gaussian case, implying their optimality.